A Course in Riemannian Geometry

David R. Wilkins

Copyright \begin{matrix}c \\ ◯ \end{matrix} David R. Wilkins 2005

Contents

1 Smooth Manifolds 3 1.1 Smooth Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Smooth Mappings between Smooth Manifolds . . . . . . . . . 5 1.4 Bump Functions and Partitions of Unity . . . . . . . . . . . . 5

2 Tangent Spaces 7 2.1 Derivatives of Smooth Maps . . . . . . . . . . . . . . . . . . . 12 2.2 Vector Fields on Smooth Manifolds . . . . . . . . . . . . . . . 12 2.3 The Lie Bracket . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Affine Connections on Smooth Manifolds 16 3.1 Vector Fields along Smooth Maps . . . . . . . . . . . . . . . . 23 3.2 Covariant Differentiation of Vector Fields along Curves …. 24 3.3 Vector Fields along Parameterized Surfaces . . . . . . . . . . . 25

4 Riemannian Manifolds 28 4.1 The Levi-Civita Connection . . . . . . . . . . . . . . . . . . . 30

5 Geometry of Surfaces in ℝ^{3} 36

6 Geodesics in Riemannian Manifolds 46 6.1 Length-Minimizing Curves in Riemannian Manifolds . . . . . 49 6.2 Geodesic Spheres and Gauss’ Lemma . . . . . . . . . . . . . . 52

7 Complete Riemannian Manifolds 55 7.1 Local Isometries and Covering Maps . . . . . . . . . . . . . . 58 8 Jacobi Fields 60

8.1 Flat Riemannian manifolds . . . . . . . . . . . . . . . . . . . . 62 8.2 The Cartan-Hadamard Theorem . . . . . . . . . . . . . . . . . 64 8.3 The Second Variation of Energy . . . . . . . . . . . . . . . . . 65

1 Smooth Manifolds

1.1 Smooth Manifolds

A topological space M is said a topological manifold of dimension n if it is metrizable (i.e., there exists a distance function d on M which generates the topology of M) and every point of M has an open neighbourhood homeomorphic to an open set in n-dimensional Euclidean space ℝ^{n}.

Let M be a topological manifold. A continuous coordinate system defined over an open set U in M is defined to be an n-tuple (x^{1}, x^{2}, …, x^{n}) of continuous real-valued functions on U such that the map φ: U → ℝ^{n} defined by

φ(u) = (x^{1}(u), x^{2}(u), …, x^{n}(u))

maps U homeomorphically onto some open set in ℝ^{n}. The domain U of the coordinate system (x^{1}, x^{2}, …, x^{n}) is referred to as a coordinate patch on M.

Two continuous coordinate systems (x^{1}, x^{2}, …, x^{n}) and (y^{1}, y^{2}, …, y^{n}) defined over coordinate patches U and V are said to be smoothly compatible if the coordinates (x^{1}, x^{2}, …, x^{n}) depend smoothly on (y^{1}, y^{2}, …, y^{n}) and vica versa on the overlap U ∩V of the coordinate patches. Note in particular that two coordinate systems are smoothly compatible if the corresponding coordinate patches are disjoint.

A smooth atlas on M is a collection of continuous coordinate systems on M such that the following two conditions hold:—

(i) every point of M belongs to the coordinate patch of at least one of these coordinate systems,

(ii) the coordinate systems in the atlas are smoothly compatible with one another.

Let A be a smooth atlas on a topological manifold M of dimension n. Let (u^{1}, u^{2}, …, u^{n}) and (v^{1}, v^{2}, …, v^{n}) be continuous coordinate systems, defined over coordinate patches U and V respectively. If the coordinate systems (u^{i}) and (v^{i}) are smoothly compatible with all the coordinate systems in the atlas A then they are smoothly compatible with each other. Indeed suppose that U ∩ V ≠ 0, and let m be a point of U ∩ V. Then (by condition (i) above) there exists a coordinate system (x^{i}) belonging to the atlas A whose coordinate patch includes that point m. But the coordinates (v^{i}) depend smoothly on the coordinates (x^{i}), and the coordinates (x^{i}) depend smoothly on the coordinates (u^{i}) around m (since the coordinate systems (u^{i}) and (v^{i}) are smoothly compatible with all coordinate systems in the atlas A). It follows from the Chain Rule that the coordinates (v^{i}) depend smoothly on the coordinates (u^{i}) around m, and similarly the coordinates (u^{i}) depend smoothly on the coordinates (v^{i}). Therefore the continuous coordinate systems (u^{i}) and (v^{i}) are smoothly compatible with each other.

We deduce that, given a smooth atlas A on a topological manifold M, we can enlarge A by adding to to A all continuous coordinate systems on M that are smoothly compatible with each of the coordinate systems of A. In this way we obtain a smooth atlas on M which is maximal in the sense that any coordinate system smoothly compatible with all the coordinate systems in the atlas already belongs to the atlas.

Definition A smooth manifold (M, A) consists of a topological manifold M together with a maximal smooth atlas A of coordinate systems on M. A smooth coordinate system (x^{1}, x^{2}, …, x^{n}) on M is a coordinate system belonging to the maximal smooth atlas A.

Note that ℝ^{n} is a smooth manifold of dimension n. The maximal smooth atlas on ℝ^{n} consists of all (curvilinear) coordinate systems that are smoothly compatible with the standard Cartesian coordinate system on ℝ^{n}.

1.2 Submanifolds

Let M be a subset of a k-dimensional smooth manifold N. We say that M is a smooth embedded submanifold of N of dimension n if, given any point m of M, there exists a smooth coordinate system (u^{1}, u^{2}, …, u^{k}) defined over some open set U in N, where m ∈ U, with the property that

M ∩ U = {p ∈ U : u^{i}(p) = 0 for i = n + 1, …, k}.

Given such a coordinate system (u^{1}, u^{2}, …, u^{k}), the restrictions of the coordinate functions u^{1}, u^{2}, …, u^{n} to U ∩ M provide a coordinate system on M around the point m. The collection of all such coordinate systems constitutes a smooth atlas on M. Thus any smooth embedded submanifold M of a smooth manifold N is itself a smooth manifold (with respect to the unique maximal smooth atlas containing the smooth atlas on M just described).

Example Consider the unit sphere S^{n} in ℝ^{n+1} consisting of those vectors x in ℝ^{n+1} satisfying |x| = 1. Given any integer i between 1 and n + 1, let

u^{j}(x) = \{ \begin{matrix}x^{j} & if j < i; \\ x^{j+1} & if i ≤ j ≤ n; \\ (x^{1})^{2} + (x^{2})^{2} + ⋯ + (x^{n+1})^{2} − 1 & if j = n + 1. \end{matrix}

Then (u^{1}, u^{2}, …, u^{n+1}) is a smooth coordinate system on U_{i}^{+} and on U_{i}^{−}, where

U_{i}^{+} = {x ∈ ℝ^{n+1} : x^{i} > 0}, U_{i}^{−} = {x ∈ ℝ^{n+1} : x^{i} < 0},

and

S^{n} ∩ U_{i}^{±} = {x ∈ U_{i}^{±} : u^{n+1}(x) = 0}.

Moreover S^{n} is covered by U_{1}^{±}, U_{2}^{±}, …, U_{n+1}^{±}. This shows that S^{n} is a smooth embedded submanifold of ℝ^{n+1}.

1.3 Smooth Mappings between Smooth Manifolds

Let M and N be smooth manifolds of dimension n and k respectively. A mapping φ: M → N from M to N is said to be smooth around a point m of M if, given smooth coordinate systems (x^{1}, x^{2}, …, x^{n}) and (y^{1}, y^{2}, …, y^{k}) around m and φ(m), the coordinates (y^{1}(φ(u)), y^{2}(φ(u)), …, y^{k}(φ(u)) of φ(u) depend smoothly on the coordinates (x^{1}(u), x^{2}(u), …, x^{n}(u)) of u for all points u belonging to some sufficiently small neighbourhood of m. (Note that if there exist smooth coordinate systems (x^{1}, x^{2}, …, x^{n}) and (y^{1}, y^{2}, …, y^{k}) around m and φ(m) for which this condition is satisfied, then the condition is satisfied for all such smooth coordinate systems around m and φ(m); this follows easily from the fact that a composition of smooth functions is smooth.) The mapping φ: M → N is said to be smooth if it is smooth around every point of M.

1.4 Bump Functions and Partitions of Unity

Let f: X → ℝ be a real-valued function defined defined over a topological space X. The support supp f of f is defined to be the closure of the set {x ∈ X : f(x) ≠ 0}. Thus supp f is the smallest closed set in X with the property that the function f vanishes on the complement of that set.

Lemma 1.1 Let U be an open set in a smooth manifold M of dimension n, and let m be a point of M. Then there exists an open subset V of U containing the point m and a smooth non-negative function f: M → ℝ such that supp f ⊂ U and f(v) = 1 for all v ∈ V.

Proof We may assume, without loss of generality, that U is contained in the coordinate patch of some smooth coordinate system (x^{1}, x^{2}, …, x^{n}) and that x^{i}(m) = 0 for i = 1, 2, …, n. Thus it suffices to show that, given any r > 0, there exists a smooth function f: ℝ^{n} → ℝ, taking values in the interval [0, 1], such that f(x) = 1 if |x| ≤ r and f(x) = 0 if |x| ≥ 2r. Let

g(t) = \{ \begin{matrix}exp(\frac{−1}{(t − 1)(4 − t)}) & if 1 < t < 4, \\ 0 & if t ≤ 1 or t ≥ 4. \end{matrix}

Then g: ℝ → ℝ is a smooth non-negative function. Indeed g(t) = h(3t − 3)h(12 − 3t) for all t, where h: ℝ → ℝ is the smooth function defined by

h(t) = \{ \begin{matrix}exp(−1/t) & if t > 0; \\ 0 & if t ≤ 0. \end{matrix}

Thus if we define

f(x) = C ∫_{|x|^{2}/r^{2}}^{+∞} g(t) dt,

where 1/C = ∫_{0}^{+∞} g(t) dt, then the function f: ℝ^{n} → ℝ has the required properties.

We now prove an existence theorem for finite ‘partitions of unity’ on a compact smooth manifold.

Theorem 1.2 Let M be a compact smooth manifold, and let V be an open cover of M. Then there exist smooth non-negative functions f_{1}, f_{2}, …, f_{k} on M with the following properties:—

(i) f_{1} + f_{2} + ⋯ + f_{k} = 1,

(ii) for each function f_{i} there exists an open set V belonging to V such that supp f_{i} ⊂ V.

Proof For each point m of M there exists a smooth non-negative function g_{m}: M → ℝ such that g_{m}(m) = 1 and supp g_{m} ⊂ V for at least one open set V belonging to V (Lemma 1.1). For each m ∈ M, let

W_{m} = \{x ∈ M : g_{m}(x) > \frac{1}{2}\}.

Then {W_{m} : m ∈ M} is an open cover of M. It follows from the compactness of M that there exists a finite collection m_{1}, m_{2}, …, m_{k} of points of M such that

M = W_{m_{1}} ∪ W_{m_{2}} ∪ ⋯ ∪ W_{m_{k}}.

Set f_{i}(x) = g_{m_{i}}(x)/G(x) for all x ∈ M, where

G(x) = g_{m_{1}}(x) + g_{m_{2}}(x) + ⋯ g_{m_{k}}(x).

Then f_{1}, f_{2}, …, f_{k} is a collection of smooth functions on M with the required properties.

A collection f_{1}, f_{2}, …, f_{k} of functions with the properties stated in Theorem 1.2 is referred to as a finite partition of unity subordinate to the open cover V of the manifold M.

Suppose that M is a (not necessarily compact) smooth manifold. A locally finite partition of unity on M is a collection of smooth non-negative functions such that

• each point of M has an open neighbourhood on which at most finitely many of the functions are non-zero,

• the sum of the values of the functions at any point of M equals 1.

Using the metrizability of M, it can be shown that there exists a locally finite partition of unity subordinate to any open cover of M.

2 Tangent Spaces

Let M be a smooth manifold of dimension n, and let m be a point of M. We say that a real-valued function f is defined around m if f is defined throughout some open neighbourhood of m. A tangent vector X_{m} at the point m can be regarded as an operator, associating a real number X_{m}[f] to any smooth real-valued function f defined around m, where

(i) X_{m}[αf + βg] = αX_{m}[f] + βX_{m}[g] for all real numbers α and β and smooth functions f and g defined around m,

(ii) X_{m}[f.g] = X_{m}[f] g(m) + f(m) X_{m}[g] for all smooth functions f and g defined around m,

(iii) if f and g are smooth real-valued functions defined around m and if f = g on some open set V containing the point m then X_{m}[f] = X_{m}[g].

(Here f.g denotes the product of the functions f and g, defined by (f.g)(m) = f(m)g(m) for all m ∈ M.) The quantity X_{m}[f] is referred to as the directional derivative of the function f along the vector X_{m}.

If X_{m} and Y_{m} are tangent vectors at the point m then, for any real numbers α and β, αX_{m} + βY_{m} is also a tangent vector at the point m, where (αX_{m} + βY_{m})[f] = αX_{m}[f] + βY_{m}[f] for all smooth real-valued functions f defined around m. It follows that the collection of all tangent vectors at the point m is a vector space T_{m}M, referred to as the tangent space to M at the point m.

Lemma 2.1 Let M be a smooth manifold, and let X_{m} be a tangent vector at some point m of M. let c_{λ} denote the constant function on M with value λ. Then X_{m}[c_{λ}] = 0 for all λ ∈ ℝ.

Proof It follows from condition (ii) above that

X_{m}[c_{1}] = X_{m}[c_{1}.c_{1}] = 2c_{1}(m)X_{m}[c_{1}] = 2X_{m}[c_{1}],

where c_{1} is the constant function with value 1. Therefore X_{m}[c_{1}] = 0, and hence X_{m}[c_{λ}] = λX_{m}[c_{1}] = 0 for all λ ∈ ℝ.

Let (x^{1}, x^{2}, …, x^{n}) be a smooth coordinate system defined over an open set U in M. Then the mapping φ: U → ℝ^{n} which sends a point u of U to (x^{1}(u), x^{2}(u), …, x^{n}(u)) maps the coordinate patch U homeomorphically onto an open set φ(U) in ℝ^{n}. The inverse φ^{−1} of φ is thus a well-defined smooth map on the image φ(U) of U.

Let m be a point of the coordinate patch U. Given any smooth real-valued function f defined around m, we denote by

\frac{∂f}{∂x^{i}} |_{m}

the ith partial derivative of the function f with respect to the coordinate system (x^{1}, x^{2}, …, x^{n}), defined by

\frac{∂f}{∂x^{i}} |_{m} = \frac{∂(f ∘ φ^{−1})}{∂t^{i}} |_{(t^{1},...,t^{n})=φ(m)} ,

where (t^{1}, t^{2}, …, t^{n}) is the standard Cartesian coordinate system on ℝ^{n}.

Given any real numbers a^{1}, a^{2}, …, a^{n}, the operator sending any smooth real-valued function f defined around the point m to

a^{1} \frac{∂f}{∂x^{1}} |_{m} + a^{2} \frac{∂f}{∂x^{2}} |_{m} + ⋯ + a^{n} \frac{∂f}{∂x^{n}} |_{m}

satisfies conditions (i)–(iii) and therefore represents a tangent vector at m which we denote by

a^{1} \frac{∂}{∂x^{1}} |_{m} + a^{2} \frac{∂}{∂x^{2}} |_{m} + ⋯ + a^{n} \frac{∂}{∂x^{n}} |_{m} .

Conversely, we shall show that any tangent vector at m is of this form for suitable real numbers a^{1}, …, a^{n}. The following lemma is is the basic result needed to prove this fact.

Lemma 2.2 Let M be a smooth manifold of dimension n and let m be a point of M. Let f be a smooth function defined over some neighbourhood of the point m. Let (x^{1}, x^{2}, …, x^{n}) be a smooth coordinate system defined around the point m. Then there exist smooth functions g_{1}, g_{2}, … g_{n}, defined over some suitable open set U containing the point m, such that

f(u) = f(m) + ∑_{i=1}^{n} (x^{i}(u) − x^{i}(m)) g_{i}(u)

for all u ∈ U. Moreover

g_{i}(m) = \frac{∂f}{∂x^{i}} |_{m} for i = 1, 2, …, n.

Proof Without loss of generality, we may assume that f is a real-valued function defined over some open ball B about the origin in ℝ^{n}. We must show that there exist smooth real-valued functions g_{1}, g_{2}, …, g_{n} on B such that

f(x) = f(0) + x^{1}g_{1}(x) + x^{2}g_{2}(x) + ⋯ + x^{n}g_{n}(x)

for all x ∈ B. Now

f(x) − f(0) = ∫_{0}^{1} \frac{d}{dt} (f(tx)) dt = ∑_{i=1}^{n} x^{i} ∫_{0}^{1}(∂_{i}f)(tx) dt,

Let

g_{i}(x) = ∫_{0}^{1}(∂_{i}f)(tx) dt

for i = 1, 2, …, n. Then g_{1}, g_{2}, …, g_{n} satisfy the required conditions.

Proposition 2.3 Let M be a smooth manifold of dimension n, and let X_{m} be a tangent vector at some point m of M. Let (x^{1}, x^{2}, …, x^{n}) be a smooth coordinate system around the point m. Then

X_{m} = a^{1} \frac{∂}{∂x^{1}} |_{m} + a^{2} \frac{∂}{∂x^{2}} |_{m} + ⋯ + a^{n} \frac{∂}{∂x^{n}} |_{m} .

where a^{i} = X_{m}[x^{i}]. If (y^{1}, y^{2}, …, y^{n}) is another smooth coordinate system around m then

X_{m} = b^{1} \frac{∂}{∂y^{1}} |_{m} + b^{2} \frac{∂}{∂y^{2}} |_{m} + ⋯ + b^{n} \frac{∂}{∂y^{n}} |_{m} ,

where

b^{j} = ∑_{i=1}^{n} a^{i} \frac{∂y^{j}}{∂x^{i}} |_{m} (j = 1, 2, …, n).

Proof Let f be a smooth real-valued function defined around m. It follows from Lemma 2.2 that there exist smooth functions g_{1}, g_{2}, …, g_{n} defined around m such that

f(u) = f(m) + ∑_{i=1}^{n} (x^{i}(u) − x^{i}(m)) g_{i}(u)

for all points u belonging to some sufficiently small open set containing m. Moreover

g_{i}(m) = \frac{∂f}{∂x^{i}} |_{m} for i = 1, 2, …, n.

Let p^{i}(u) = x^{i}(u) − x^{i}(m). Now the operator X_{m} annihilates constant functions, by Lemma 2.1. Therefore X_{m}[p^{i}] = X_{m}[x^{i}] = a^{i} for all i, and hence

X_{m}[f] = ∑_{i=1}^{n} (X_{m}[p^{i}] g_{i}(m) + p^{i}(m) X_{m}[g^{i}]) = ∑_{i=1}^{n} a^{i} \frac{∂f}{∂x^{i}} |_{m} .

If (y^{1}, y^{2}, …, y^{n}) is another smooth coordinate system around m then

\frac{∂f}{∂x^{i}} = ∑_{j=1}^{n} \frac{∂y^{j}}{∂x^{i}} \frac{∂f}{∂y^{j}} ,

by the Chain Rule, and hence

X_{m} = b^{1} \frac{∂}{∂y^{1}} |_{m} + b^{2} \frac{∂}{∂y^{2}} |_{m} + ⋯ + b^{n} \frac{∂}{∂y^{n}} |_{m} ,

where

b^{j} = ∑_{i=1}^{n} a^{i} \frac{∂y^{j}}{∂x^{i}} |_{m} (j = 1, 2, …, n),

as required.

Corollary 2.4 Let M be a smooth manifold of dimension n. Then the tangent space T_{m}M to M at any point m of M has dimension n. Moreover, given any smooth coordinate system (x^{1}, x^{2}, …, x^{n}) around m, the tangent vectors

\frac{∂}{∂x^{1}} |_{m} , \frac{∂}{∂x^{2}} |_{m} , …, \frac{∂}{∂x^{n}} |_{m}

constitute a basis for the tangent space T_{m}M.

Proof It follows immediately from Proposition 2.3 that these tangent vectors span the tangent space T_{m}M. It thus suffices to show that they are linearly independent. Suppose that

∑_{i=1}^{n} a^{i} \frac{∂}{∂x^{i}} |_{m} = 0

for some real numbers a^{1}, a^{2}, …, a^{n}. Then

a^{j} = (∑_{i=1}^{n} a^{i} \frac{∂}{∂x^{i}} |_{m}) (x^{j}) = 0

for j = 1, 2, …, n, showing that the tangent vectors ∂/∂x^{i} (i = 1, 2, …, n) are linearly independent at m, as required.

Let γ: I → M be a smooth curve in the smooth manifold M, where I is some interval in ℝ. Then γ determines, for each t ∈ I, a tangent vector γ′(t) at the point γ(t), defined by

γ′(t)[f] = \frac{df(γ(t))}{dt}.

We refer to the tangent vector γ′(t) as the velocity vector of the curve γ at γ(t).

Every tangent vector at a point m of the smooth manifold M is the velocity vector of some smooth curve passing through the point m. Indeed let (x^{1}, x^{2}, …, x^{n}) be a smooth coordinate system around the point m chosen such that x^{i}(m) = 0 for i = 1, 2, …, n. Let X_{m} be a tangent vector at the point m. Then

X_{m} = a^{1} \frac{∂}{∂x^{1}} |_{m} + a^{2} \frac{∂}{∂x^{2}} |_{m} + ⋯ + a^{n} \frac{∂}{∂x^{n}} |_{m}

bu Proposition 2.3, where a^{i} = X_{m}[x^{i}] for i = 1, 2, …, n. Let γ: (−ε, ε) → M be the smooth curve in M given by x^{i}(γ(t)) = a^{i}t for i = 1, 2, …, n (where ε is some suitably small positive real number). It follows from the Chain Rule that

γ′(0)[f] = \frac{df(γ(t))}{dt}|_{t=0} = ∑_{i=1}^{n} \frac{∂f}{∂x^{i}} |_{m} \frac{d(x^{i}(γ(t))}{dt}|_{t=0} = ∑_{i=1}^{n} a^{i} \frac{∂f}{∂x^{i}} |_{m} = X_{m}[f]

for all smooth real-valued functions f defined around m, so that γ′(0) = X_{m}.

2.1 Derivatives of Smooth Maps

Let φ: M → N be a smooth map between smooth manifolds of dimensions n and k respectively. Let X_{m} be a tangent vector at some point m of M. Given any smooth function f, defined around φ(m) in N, we define (φ_{∗}X_{m})[f] = X_{m}[f ∘ φ].

Lemma 2.5 The operator φ_{∗}X_{m} is a tangent vector at φ(m).

Proof Let f and g be smooth real-valued functions defined around φ(m), and let α and β be real numbers. Then

(φ_{∗}X_{m})[αf + βg] = X_{m}[α(f ∘ φ) + β(g ∘ φ)] = αX_{m}[f ∘ φ] + βX_{m}[g ∘ φ]
= α(φ_{∗}X_{m})[f] + β(φ_{∗}X_{m})[g],
(φ_{∗}X_{m})[f.g] = X_{m}[f ∘ φ]g(φ(m)) + f(φ(m))X_{m}[g ∘ φ]
= (φ_{∗}X_{m}[f]g(φ(m)) + f(φ(m))(φ_{∗}X_{m})[g]

Moreover if the functions f and g agree on some open set containing φ(m) then the functions f ∘φ and g ∘φ agree on some open set containing m (since φ: M → N is continuous), and therefore (φ_{∗}X_{m})[f] = (φ_{∗}X_{m})[g]. Thus the operator φ_{∗}X_{m} is a tangent vector at φ(m).

Given a smooth map φ: M → N between smooth manifolds M and N, the linear transformation φ_{∗}: T_{m}M → T_{φ(m)}N between the tangent spaces at m and φ(m) is referred to as the derivative of the smooth map φ at the point m.

Let (x^{1}, …, x^{n}) and (y^{1}, …, y^{k}) be smooth coordinate systems on M and N respectively around the points m and φ(m). Then the map φ is determined around m by smooth functions F^{1}, F^{2}, …, F^{k}, where

F^{j}(x^{1}, x^{2}, …, x^{n}) = y^{j} ∘ φ (j = 1, 2, …, k).

A straightforward application of the Chain Rule for functions of real variables shows that

φ_{∗} (∑_{i=1}^{n} a^{i} \frac{∂}{∂x^{i}} |_{m}) = ∑_{j=1}^{k} (∑_{i=1}^{n} a^{i} \frac{∂F^{j}}{∂x^{i}} |_{m}) \frac{∂}{∂y^{j}} |_{φ(m)} .

2.2 Vector Fields on Smooth Manifolds

Given any open set U in a smooth manifold M we denote by C^{∞}(U) the algebra of all smooth real-valued functions on U.

Definition Let U be an open set in a smooth manifold M. A smooth vector field X on U is an operator X: C^{∞}(U) → C^{∞}(U), sending f ∈ C^{∞}(U) to X[f], where

X[f + h] = X[f] + X[h], X[λf] = λX[f] and X[f.h] = X[f].h + f.X[h]

for all f, h ∈ C^{∞}(U) and λ ∈ ℝ.

If X is a smooth vector field on U then X[c] = 0 for all constant functions c on U.

Lemma 2.6 Let X be a smooth vector field on an open set U in a smooth manifold M, and let V be an open subset of U. Let f and h be smooth realvalued functions on U. Suppose that f = h on V. Then X[f] = X[h] on V.

Proof Let m be a point of V. Then there exists a smooth bump function β: U → ℝ such that β(m) = 1 and β(u) = 0 for all u ∈ U \ V (see Lemma 1.1). Then β.(f − h) = 0, hence

0 = X[β.(f − h)] = X[β].(f − h) + β. (X[f] − X[h])

But f(m) − h(m) = 0 and β(m) = 1. Therefore X[f](m) = X[h](m), as required.

Let f be a smooth real-valued function defined around the point m. Then there exists a smooth function f̃: M → ℝ on M such that f = f̃ throughout some open set V containing the point m. Indeed we can take

f̃(p) = \{ \begin{matrix}β(p)f(p) & if p ∈ supp β, \\ 0 & if p ∉ supp β, \end{matrix}

where β: M → ℝ is a smooth bump function with the following properties:

• β(v) = 1 for all points v of some open set V containing the point m,

• the support supp β of β is contained in the domain of the function f.

Given a smooth vector field X defined over some open set U in M, and let m be a point of U. Given any smooth real-valued function f defined around m, we define X_{m}[f] = X[f̃](m), where f̃: M → ℝ is some smooth function on M with the property that f̃ = f over some open set V containing the point m. It follows from Lemma 2.6 that the value of X_{m}[f] does not depend on the choice of f̃. Moreover the operator X_{m} sending any smooth realvalued function f defined around m to the real number X_{m}[f] is a tangent vector at the point m. Moreover X_{m}[f] = X[f](m) for all f ∈ C^{∞}(U). We refer to X_{m} as the value of the vector field X at the point m.

Let (x^{1}, x^{2}, …, x^{n}) be a smooth coordinate system defined over some open set U in M. Then the operator ∂/∂x^{i} sending any smooth function f: U → ℝ to ∂f/∂x^{i} is a smooth vector field on U for i = 1, 2, …, n. Thus, given any smooth real-valued functions a^{1}, a^{2}, …, a^{m} on U, the operator

a^{1} \frac{∂}{∂x^{1}} + a^{2} \frac{∂}{∂x^{2}} + ⋯ + a^{n} \frac{∂}{∂x^{n}}

is a smooth vector field on U. The following result now follows directly using Proposition 2.3.

Proposition 2.7 Let M be a smooth manifold of dimension n, and let X be a smooth vector field defined over the domain U of some smooth coordinate system (x^{1}, x^{2}, …, x^{n}) on M. Then

X = a^{1} \frac{∂}{∂x^{1}} + a^{2} \frac{∂}{∂x^{2}} + ⋯ + a^{n} \frac{∂}{∂x^{n}}

where a^{1}, a^{2}, …, a^{n} are the smooth functions on U given by a^{i} = X[x^{i}], for i = 1, 2, …, n.

2.3 The Lie Bracket

Let M be a smooth manifold and let X and Y be smooth vector fields on some open set U in M. We denote by [X, Y] the operator on C^{∞}(U) defined by

[X, Y][f] = X[Y [f]] − Y [X[f]]

for all smooth real-valued functions f on U. Now

[X, Y][f + g] = [X, Y][f] + [X, Y][g], [X, Y][λf] = λ[X, Y][f]

for all f, g ∈ C^{∞}(U) and λ ∈ ℝ. Also

X[Y [f.g]] = X[(Y [f].g + f.Y [g])]
= X[Y [f]].g + Y [f].X[g] + X[f].Y [g] + f.X[Y [g]],

and hence

[X, Y][f.g] = X[Y [f.g]] − Y [X[f.g]]
= X[Y [f]].g − Y [X[f].g + f.X[Y [g]] − f.Y [X[g]]
= [X, Y][f].g + f.[X, Y][g].

We deduce that the operator [X, Y] on C^{∞}(U) is a smooth vector field on U. This vector field is referred to as the Lie bracket of the vector fields X and Y on U.

Note that if X is a smooth vector field on U, then so is fX for any f ∈ C^{∞}(U), where fX[g] = f.X[g] for all g ∈ C^{∞}(U).

Lemma 2.8 Let M be a smooth manifold. Let X and Y be smooth vector fields on M and let f and g be smooth real-valued functions on M. Then

[fX, gY] = (f.g)[X, Y] + (f.X[g])Y − (g.Y [f])X.

Proof Let h be a smooth real-valued function defined over some open set in M. Then

[fX, gY][h] = f.X[g.Y [h]] − g.Y [f.X[h]]
= (f.g).X[Y [h]] + f.X[g].Y [h] − (f.g).Y [X[h]] − g.Y [f].X[h]
= ((f.g)[X, Y] + (f.X[g])Y − (g.Y [f])X) [h],

as required.

Lemma 2.9 Let M be a smooth manifold of dimension n and let X and Y be smooth vector fields on some open set U in M. Let (x^{1}, x^{2}, …, x^{n}) be a smooth coordinate system on U. Suppose that

X = ∑_{i=1}^{n} u^{i} \frac{∂}{∂x^{i}} , Y = ∑_{i=1}^{n} v^{i} \frac{∂}{∂x^{i}} .

Then

[X, Y] = ∑_{i=1}^{n} ∑_{j=1}^{n} (u^{j} \frac{∂v^{i}}{∂x^{j}} − v^{j} \frac{∂u^{i}}{∂x^{j}} ) \frac{∂}{∂x^{i}}

Proof Let f ∈ C^{∞}(U). Then

[X, Y][f] = X [∑_{i=1}^{n} v^{i} \frac{∂f}{∂x^{i}} ] − Y [∑_{i=1}^{n} u^{i} \frac{∂f}{∂x^{i}} ]

= ∑_{i=1}^{n} ∑_{j=1}^{n} (u^{j} \frac{∂}{∂x^{j}} (v^{i} \frac{∂f}{∂x^{i}} ) − v^{j} \frac{∂}{∂x^{j}} (u^{i} \frac{∂f}{∂x^{i}} ))

= ∑_{i=1}^{n} ∑_{j=1}^{n} (u^{j} \frac{∂v^{i}}{∂x^{j}} − v^{j} \frac{∂u^{i}}{∂x^{j}} ) \frac{∂f}{∂x^{i}} ,

since

∑_{i=1}^{n} ∑_{j=1}^{n}(u^{j}v^{i} − v^{j}u^{i})\frac{∂^{2}f}{∂x^{j}∂x^{i}} = ∑_{i=1}^{n} ∑_{j=1}^{n} u^{j}v^{i} (\frac{∂^{2}f}{∂x^{j}∂x^{i}} − \frac{∂^{2}f}{∂x^{i}∂x^{j}} ) = 0.

Note in particular that, for any smooth coordinate system (x^{1}, x^{2}, …, x^{n}) on M,

[\frac{∂}{∂x^{i}} , \frac{∂}{∂x^{j}} ] = 0 (i, j = 1, 2, …, n).

Let M and N be smooth manifolds and let φ: M → N be a smooth map from M to N. Let X and X̃ be smooth vector fields on M and N respectively. We say that X and X̃ are φ-related if X̃[g] ∘ φ = X[g ∘ φ] for all smooth real-valued functions g on N.

Lemma 2.10 Let M and N be smooth manifolds and let φ: M → N be a smooth map from M to N. Let X and Y be smooth vector fields on M, and let X̃ and Ỹ be smooth vector fields on N. Suppose that the vector fields X and X̃ are φ-related and that the vector fields Y and Ỹ are φ-related. Then the vector fields [X, Y] and [X̃, Ỹ] are also φ-related.

Proof Let g ∈ C^{∞}(N). Then

[X̃, Ỹ][g] ∘ φ = X̃[Ỹ [g]] ∘ φ − Ỹ [X̃[g]] ∘ φ = X[Ỹ [g] ∘ φ] − Y [X̃[g] ∘ φ]
= X[Y [g ∘ φ]] − Y [X[g ∘ φ]] = [X, Y][g ∘ φ],

as required.

3 Affine Connections on Smooth Manifolds

Definition Let M be a smooth manifold. An affine connection on M is a differential operator, sending smooth vector fields X and Y to a smooth vector field ∇_{X}Y, which satisfies the following conditions:

\begin{matrix}∇_{X+Y} Z = ∇_{X}Z + ∇_{Y} Z, & ∇_{X}(Y + Z) = ∇_{X}Y + ∇_{X}Z, \\ ∇_{fX}Y = f ∇_{X}Y, & ∇_{X}(fY) = X[f] Y + f ∇_{X}Y \end{matrix}

for all smooth vector fields X, Y and Z and real-valued functions f on M. The vector field ∇_{X}Y is known as the covariant derivative of the vector field Y along X (with respect to the affine connection ∇). The torsion tensor T and the curvature tensor R of an affine connection ∇ are the operators sending smooth vector fields X, Y and Z on M to the smooth vector fields T (X, Y) and R(X, Y)Z given by

T (X, Y) = ∇_{X}Y − ∇_{Y} X − [X, Y],
R(X, Y)Z = ∇_{X}∇_{Y} Z − ∇_{Y} ∇_{X}Z − ∇_{[X,Y]}Z.

An affine connection ∇ on M is said to be torsion-free if its torsion tensor is everywhere zero (so that ∇_{X}Y − ∇_{Y} X = [X, Y] for all smooth vector fields X and Y on M).

Example Let U be an open set in ℝ^{n}, and let X and Y be smooth vector fields on U. Then

X = ∑_{i=1}^{n} a^{i} \frac{∂}{∂x^{i}} , Y = ∑_{i=1}^{n} b^{i} \frac{∂}{∂x^{i}} ,

where a^{1}, a^{2}, …, a^{n} and b^{1}, b^{2}, …, b^{n} are the components of the vector fields X and Y with respect to the Cartesian coordinate system (x^{1}, x^{2}, …, x^{n}) on ℝ^{n}. The directional derivative ∂_{X}Y of the vector field Y along the vector field X is then given by the formula

∂_{X}Y = ∑_{i=1}^{n} X[b^{i}]\frac{∂}{∂x^{i}} = ∑_{i,j=1}^{n} a^{j} \frac{∂b^{i}}{∂x^{j}} \frac{∂}{∂x^{i}}

(where X[b^{i}] denotes the directional derivative of the function b^{i} along the vector field X). Then the differential operator sending smooth vector fields X and Y to ∂_{X}Y is an affine connection on U. We refer to this affine connection as the canonical (or usual) flat connection on the open set U. Now

∂_{X}Y − ∂_{Y} X = ∑_{i=1}^{n} ∑_{j=1}^{n} (a^{j} \frac{∂b^{i}}{∂x^{j}} − b^{j} \frac{∂a^{i}}{∂x^{j}} ) \frac{∂}{∂x^{i}} = [X, Y]

(see Lemma 2.9). Thus the canonical flat connection ∂ on U is torsion-free. Moreover, given any smooth vector field Z on U with Cartesian compoents c^{1}, c^{2}, …, c^{n}, we see that

∂_{X}∂_{Y} Z − ∂_{Y} ∂_{X}Z = (∂_{X}∂_{Y} − ∂_{Y} ∂_{X}) (∑_{i=1}^{n} c^{i} \frac{∂}{∂x^{i}} )

= ∑_{i=1}^{n} (X[Y [c^{i}]] − Y [X[c^{i}]]) \frac{∂}{∂x^{i}} = ∑_{i=1}^{n}[X, Y][c^{i}]\frac{∂}{∂x^{i}}
= ∂_{[X,Y]}Z.

We deduce that the curvature tensor of the canonical flat connection ∂ on U is zero everywhere on U.

Example Let M be a smooth n-dimensional submanifold of ℝ^{k}. This means that M is a subset of ℝ^{k} with the property that, for all m ∈ M, there exists a smooth (curvilinear) coordinate system (u^{1}, u^{2}, …, u^{k}) defined over a neighbourhood U of m which has the property that

M ∩ U = {p ∈ U : u^{j}(p) = 0 for j = n + 1, …, k}.

Let X and Y be smooth vector fields on M that are everywhere tangent to M, and let a^{1}, a^{2}, …, a^{n} and b^{1}, b^{2}, …, b^{n} be the components of X and Y with respect to the curvilinear coordinate system (u^{1}, u^{2}, …, u^{n}) on M. Thus

X = ∑_{j=1}^{n} a^{j} \frac{∂}{∂u^{j}} , Y = ∑_{j=1}^{n} b^{j} \frac{∂}{∂u^{j}} = ∑_{i=1}^{k} c^{i} \frac{∂}{∂x^{i}} ,

where (x^{1}, x^{2}, …, x^{k}) is the standard Cartesian coordinate system of ℝ^{k}, and

c^{i} = ∑_{j=1}^{n} b^{j} \frac{∂x^{i}}{∂u^{j}} (i = 1, 2, …, k).

The directional derivative ∂_{X}Y of the vector field Y along X at each point of M is defined to be the vector in ℝ^{k} whose Cartesian components are the directional derivatives X[c^{1}], X[c^{2}], …, X[c^{k}] of the Cartesian components c^{1}, c^{2}, …, c^{k} of Y along the vector field X. Thus

∂_{X}Y = ∑_{i=1}^{k} X[c^{i}]\frac{∂}{∂x^{i}} = ∑_{i=1}^{k} ∑_{j,l=1}^{n} a^{j} \frac{∂}{∂u^{j}} (b^{l} \frac{∂x^{i}}{∂u^{l}} ) \frac{∂}{∂x^{i}}

= ∑_{i=1}^{k} ∑_{j,l=1}^{n} (a^{j} \frac{∂b^{l}}{∂u^{j}} \frac{∂x^{i}}{∂u^{l}} + a^{j}b^{l} \frac{∂^{2}x^{i}}{∂u^{j}∂u^{l}} ) \frac{∂}{∂x^{i}} .

Now ∂_{X}Y is not necessarily tangential to to the submanifold M at each point of M. However ∂_{X}Y − ∂_{Y} X is always tangential to M, and ∂_{X}Y − ∂_{Y} X = [X, Y]. Indeed

∂_{X}Y − ∂_{Y} X = ∑_{i=1}^{k} ∑_{j,l=1}^{n} (a^{j} \frac{∂b^{l}}{∂u^{j}} − b^{j} \frac{∂a^{l}}{∂u^{j}} ) \frac{∂x^{i}}{∂u^{l}} \frac{∂}{∂x^{i}}

= ∑_{j,l=1}^{n} (a^{j} \frac{∂b^{l}}{∂u^{j}} − b^{j} \frac{∂a^{l}}{∂u^{j}} ) \frac{∂}{∂u^{l}}
= [X, Y].

We split ∂_{X}Y into its tangential and normal components, writing

∂_{X}Y = ∇_{X}Y + S(X, Y),

where ∇_{X}Y is everywhere tangential to M and S(X, Y) is everywhere normal to M (i.e., at each point m of M, the vector S(X, Y) is orthogonal to the tangent space T_{m}M to M at the point m). Now

∂_{X_{1}+X_{2}}Y = ∂_{X_{1}}Y + ∂_{X_{2}}Y, ∂_{X}(Y_{1} + Y_{2}) = ∂_{X}Y_{1} + ∂_{X}Y_{2}

∂_{fX}Y = f ∂_{X}Y, ∂_{X}(fY) = f ∂_{X}Y + X[f] Y

for all smooth real-valued functions f and (tangential) vector fields X, X_{1}, X_{2}, Y, Y_{1}, Y_{2} on M. On splitting these identities into their tangential and normal components, we deduce that the differential operator ∇ is an affine connection on M, and

S(X_{1} + X_{2}, Y) = S(X_{1}, Y) + S(X_{2}, Y),
S(X, Y_{1} + Y_{2}) = S(X, Y_{1}) + S(X, Y_{2}),
S(fX, Y) = fS(X, Y) = S(X, fY).

Moreover

∇_{X}Y − ∇_{Y} X = [X, Y], S(X, Y) = S(Y, X),

since ∂_{X}Y − ∂_{Y} X = [X, Y]. Thus the affine connection ∇ is torsion-free.

Lemma 3.1 Let ∇ be an affine connection on a smooth manifold M. Then the value of the covariant derivative ∇_{X}Y at a point m of M depends only on the vector field Y and on the value X_{m} of the vector field X at the point m.

Proof Let (x^{1}, x^{2}, …, x^{n}) be a smooth coordinate system defined around m. Then

X = ∑_{i=1}^{n} a^{i} \frac{∂}{∂x^{i}} , Y = ∑_{i=1}^{n} b^{i} \frac{∂}{∂x^{i}} ,

where a^{1}, a^{2}, …, a^{n} and b^{1}, b^{2}, …, b^{n} are smooth real-valued functions defined around m. Now

∇\frac{∂}{∂x^{j}} (\frac{∂}{∂x^{k}} ) = ∑_{i=1}^{n} Γ_{jk}^{i} \frac{∂}{∂x^{i}} ,

where Γ_{jk}^{i} are smooth functions defined over the domain of the coordinate system (x^{1}, x^{2}, …, x^{n}). It follows from the definition of an affine connection that

∇_{X}Y = ∑_{k=1}^{n}(X[b^{k}]\frac{∂}{∂x^{k}} + b^{k}∇_{X}(\frac{∂}{∂x^{k}} ))

= ∑_{i,j=1}^{n}(a^{j} \frac{∂b^{i}}{∂x^{j}} + ∑_{k=1}^{n} a^{j}b^{k}Γ_{jk}^{i})\frac{∂}{∂x^{i}} .

Therefore the value of ∇_{X}Y at the point m depends only on the functions b^{1}, …, b^{n} and their partial derivatives at m and on the values of the functions a^{1}, a^{2}, …, a^{n} at m. We deduce that the value of ∇_{X}Y at m depends only on the vector field Y and on the value X_{m} of X at m.

Let ∇ be an affine connection on a smooth manifold M, let Y be a smooth vector field on M, and let X_{m} be a tangent vector at some point m of M. We define ∇_{X_{m}}Y to be the value of ∇_{X}Y at m, where X is any smooth vector field on M whose value at m is X_{m}. Lemma 3.1 shows that ∇_{X_{m}}Y is well-defined. We refer to ∇_{X_{m}}Y as the covariant derivative of the vector field Y along the tangent vector X_{m} at the point m.

Lemma 3.2 Let ∇ be an affine connection on a smooth manifold M and let T be the torsion tensor of the affine connection ∇. Then

T (X, Y) = −T (Y, X)

for all vector fields X and Y on M. Also

\begin{matrix}T (X + Y, Z) = T (X, Z) + T (Y, Z), & T (X, Y + Z) = T (X, Y) + T (X, Z), \\ T (fX, Y) = f T (X, Y), & T (X, fY) = f T (X, Y) \end{matrix}

for all smooth vector fields X, Y and Z on M and for all smooth real-valued functions f on M. Moreover the value of T (X, Y) at any point m of M depends only on the values X_{m} and Y_{m} of X and Y at m.

Proof The identities

T (X, Y) = −T (Y, X), T (X + Y, Z) = T (X, Z) + T (Y, Z),
T (X, Y + Z) = T (X, Y) + T (X, Z)

follow immediately from the definition of T. Now if f is a smooth real-valued function on M then [X, fY] = f [X, Y] + X[f] Y by Lemma 2.8. Thus

T (X, fY) = ∇_{X}(fY) − ∇_{fY} X − [X, fY]
= f ∇_{X}Y + X[f] Y − f ∇_{Y} X − f [X, Y] − X[f] Y
= fT (X, Y).

Also

T (fX, Y) = −T (Y, fX) = −f T (Y, X) = f T (X, Y).

Let m be a point of M and let (x^{1}, x^{2}, …, x^{n}) be a coordinate system defined around m. Then

X = ∑_{i=1}^{n} a^{i} \frac{∂}{∂x^{i}} , Y = ∑_{i=1}^{n} b^{i} \frac{∂}{∂x^{i}} ,

where a^{1}, a^{2}, …, a^{n} and b^{1}, b^{2}, …, b^{n} are smooth real-valued functions defined around m. Let T^{i}_{jk} be the real-valued functions on the domain of the coordinate system characterized by the property that

T (\frac{∂}{∂x^{j}} , \frac{∂}{∂x^{k}} ) = ∑_{i=1}^{n} T^{i}_{jk} \frac{∂}{∂x^{i}} .

It follows from the identities derived above that

T (X, Y) = ∑_{i=1}^{n} ∑_{j=1}^{n} ∑_{k=1}^{n} (T^{i}_{jk}a^{j}b^{k} \frac{∂}{∂x^{i}} ) .

This shows that the value of T (X, Y) at m depends only on the values of the functions a^{1}, a^{2}, …, a^{n} and b^{1}, b^{2}, …, b^{n} at m, and thus depends only on the values X_{m} and Y_{m} of the vector fields X and Y at m.

Lemma 3.3 Let ∇ be an affine connection on a smooth manifold M with curvature tensor R. Then

R(X, Y)Z = −R(Y, X)Z,
R(X_{1} + X_{2}, Y)Z = R(X_{1}, Y)Z + R(X_{2}, Y)Z,
R(X, Y_{1} + Y_{2})Z = R(X, Y_{1})Z + R(X, Y_{2})Z,
R(X, Y)(Z_{1} + Z_{2}) = R(X, Y)Z_{1} + R(X, Y)Z_{2},
R(fX, Y)Z = R(X, fY)Z = R(X, Y)(fZ) = f R(X, Y)Z,

for all vector fields X, X_{1}, X_{2}, Y, Y_{1}, Y_{2}, Z, Z_{1} and Z_{2} and smooth functions f on M. Moreover the value of R(X, Y)Z at any point m of M depends only on the values X_{m}, Y_{m} and Z_{m} of X, Y and Z at m.

Proof The first four identities follow directly from the definition of the curvature tensor. Let X, Y and Z be smooth vector fields on M and let f be a smooth real-valued function on M. Then

[X, fY] = f [X, Y] + X[f] Y,

hence

R(X, fY)Z = ∇_{X}∇_{fY} Z − ∇_{fY} ∇_{X}Z − ∇_{(f [X,Y]+X[f] Y)}Z
= ∇_{X}(f ∇_{Y} Z) − f ∇_{Y} ∇_{X}Z − f ∇_{[X,Y]}Z − X[f] ∇_{Y} Z
= f ∇_{X}∇_{Y} Z + X[f]∇_{Y} Z − f ∇_{Y} ∇_{X}Z − f ∇_{[X,Y]}Z
− X[f] ∇_{Y} Z
= f R(X, Y)Z.

Also

R(fX, Y)Z = −R(Y, fX)Z = −f R(Y, X)Z = f R(X, Y)Z,

and

R(X, Y)(fZ) = ∇_{X} (f ∇_{Y} Z + Y [f] Z) − ∇_{Y} (f ∇_{X}Z + X[f] Z)
− f ∇_{[X,Y]}Z − [X, Y][f] Z
= (f ∇_{X}∇_{Y} Z + X[f] ∇_{Y} Z + Y [f] ∇_{X}Z + X[Y [f]] Z)
− (f ∇_{Y} ∇_{X}Z + Y [f] ∇_{X}Z + X[f] ∇_{Y} Z + Y [X[f]] Z)
− f ∇_{[X,Y]}Z − X[Y [f]] Z + Y [X[f]] Z
= f R(X, Y)Z.

Thus

R(fX, Y)Z = R(X, fY)Z = R(X, Y)(fZ) = f R(X, Y)Z.

We now show that the value of R(X, Y)Z depends only on the values of the vector fields X, Y and Z at m. Choose a smooth coordinate system (x^{1}, x^{2}, …, x^{n}) around m. Let the functions (R^{i}_{jkl}) be defined such that

R (\frac{∂}{∂x^{k}} , \frac{∂}{∂x^{l}} ) \frac{∂}{∂x^{j}} = ∑_{i=1}^{n} R^{i}_{jkl} \frac{∂}{∂x^{i}} .

Let

X = ∑_{k=1}^{n} u^{k} \frac{∂}{∂x^{k}} ,  Y = ∑_{l=1}^{n} v^{l} \frac{∂}{∂x^{l}} ,  Z = ∑_{j=1}^{n} a^{j} \frac{∂}{∂x^{j}} .

Then

R(X, Y)Z = ∑_{i,j,k,l} (R^{i}_{jkl}a^{j}u^{k}v^{l} \frac{∂}{∂x^{i}} ) .

We deduce that the value of R(X, Y)Z depends only on the values of X, Y and Z at m, as required.

Let ∇ be an affine connection on a smooth manifold M, let Y be a smooth vector field on M, and let X_{m}, Y_{m} and Z_{m} be tangent vectors at some point m of M. We define T (X_{m}, Y_{m}) and R(X_{m}, Y_{m})Z_{m} to be the values of the vector fields T (X, Y) and R(X, Y)Z at m, where X, Y and Z are any smooth vector fields on M whose values at m are X_{m}, Y_{m} and Z_{m} respectively. Lemma 3.2 and Lemma 3.3 show that T (X_{m}, Y_{m}) and R(X_{m}, Y_{m})Z_{m} are well-defined, and do not depend on the choice of X, Y and Z.

Lemma 3.4 (The First Bianchi Identity) Let ∇ be a torsion-free affine connection on a smooth manifold M. Let R denote the curvature operator of ∇. Then

R(X, Y)Z + R(Y, Z)X + R(Z, X)Y = 0

Proof The connection ∇ is torsion-free, hence ∇_{X}Y −∇_{Y} X = [X, Y] for all vector fields X and Y on M. Therefore

R(X, Y)Z + R(Y, Z)X + R(Z, X)Y

= ∇_{X}∇_{Y} Z + ∇_{Y} ∇_{Z}X + ∇_{Z}∇_{X}Y
− ∇_{Y} ∇_{X}Z − ∇_{Z}∇_{Y} X − ∇_{X}∇_{Z}Y
− ∇_{[X,Y]}Z − ∇_{[Y,Z]}X − ∇_{[Z,X]}Y
= ∇_{X} (∇_{Y} Z − ∇_{Z}Y) + ∇_{Y} (∇_{Z}X − ∇_{X}Z) + ∇_{Z} (∇_{X}Y − ∇_{Y} Z)
− ∇_{[X,Y]}Z − ∇_{[Y,Z]}X − ∇_{[Z,X]}Y
= ∇_{X}[Y, Z] + ∇_{Y} [Z, X] + ∇_{Z}[X, Y]
− ∇_{[X,Y]}Z − ∇_{[Y,Z]}X − ∇_{[Z,X]}Y
= 0.

3.1 Vector Fields along Smooth Maps

Let P and M be smooth manifolds, and let α: P → M be a smooth map. A vector field V along the map α is defined to be a map which associates to each point p of P a tangent vector V (p) to M at α(p). If (x^{1}, x^{2}, …, x^{n}) is a smooth coordinate system defined over some open set U in M then we can write

V (p) = ∑_{i=1}^{n} c^{i}(p) \frac{∂}{∂x^{i}} |_{α(p)}

for all p ∈ α^{−1}(U). We say that a vector field V along the map α is smooth if, given any smooth coordinate system (x^{1}, x^{2}, …, x^{n}) defined over an open subset U of M, the components c^{1}, c^{2}, …, c^{n} of V with respect to this coordinate system are smooth functions on α^{−1}(U). In particular, one can define in this way smooth vector fields along curves and surfaces in the smooth manifold M.

Let γ: I → M be a smooth curve in the smooth manifold M, where I is some open interval in ℝ. Then a vector field V along the curve γ is a function which associates to t ∈ I a tangent vector V (t) to M at γ(t). In particular, the mapping sending t ∈ I to the velocity vector γ′(t) of the curve γ at γ(t) provides an example of a smooth vector field along the curve γ.

3.2 Covariant Differentiation of Vector Fields along Curves

Let M be a smooth manifold, and let ∇ be an affine connection on M. Let γ: I → M be a smooth curve in the smooth manifold M, where I is some open interval in ℝ. Given any t ∈ I, we can find smooth vector fields X_{1}, X_{2}, …, X_{n}, defined over some open set U containing γ(t), with the property that the values (X_{1})_{m}, (X_{2})_{m}, …, (X_{n})_{m} of these vector fields at any point m of U constitute a basis for the tangent space T_{m}M of M at m. (For example, we can take X_{i} = ∂/∂x^{i}, where (x^{1}, x^{2}, …, x^{n}) is any smooth coordinate system defined over U.) If V is any smooth vector field along the curve γ, we can write

V (t) = ∑_{i=1}^{n} v^{i}(t)(X_{i})_{γ(t)},

where v^{1}, v^{2}, …, v^{n} are smooth functions on γ^{−1}(U). We define the covariant derivative of V along the curve γ at γ(t) by the formula

\frac{DV (t)}{dt} = ∑_{i=1}^{n} (\frac{dv^{i}(t)}{dt}(X_{i})_{γ(t)} + v^{i}(t)∇_{γ′(t)}X_{i}) ,

where γ′(t) denotes the velocity vector of the curve γ at γ(t). We claim that this expression defining the covariant derivative does not depend on the choice of the vector fields X_{1}, X_{2}, …, X_{n} around γ(t). Indeed suppose that Y_{1}, Y_{2}, … Y_{n} are vector fields on U which form a basis of the tangent space at each point of U. Then

X_{i} = ∑_{j=1}^{n} a_{i}^{j}Y_{j}

on U, where (a_{i}^{j}) is an n × n matrix of smooth functions on U which is non-singular at each point of U. Thus

V (t) = ∑_{j=1}^{n} w^{j}(t)(Y_{j})_{γ(t)}, where w^{j}(t) = ∑_{i=1}^{n} v^{i}(t)a_{i}^{j}(γ(t))

for all t ∈ γ^{−1}(U), and

∑_{j=1}^{n} (\frac{dw^{j}(t)}{dt}(Y_{j})_{γ(t)} + w^{j}(t)∇_{γ′(t)}Y_{j})

= ∑_{i,j=1}^{n}(\frac{dv^{i}(t)}{dt}a_{i}^{j}(γ(t))(Y_{j})_{γ(t)} + v^{i}(t)\frac{d(a_{i}^{j} ∘ γ)(t)}{dt}(Y_{j})_{γ(t)}

+ v^{i}(t)a_{i}^{j}(γ(t))∇_{γ′(t)}Y_{j})
= ∑_{i=1}^{n}(\frac{dv^{i}(t)}{dt}(X_{i})_{γ(t)} + v^{i}(t)∇_{γ′(t)}(∑_{j=1}^{n} a_{i}^{j}Y_{j}))

= ∑_{i=1}^{n} (\frac{dv^{i}(t)}{dt}(X_{i})_{γ(t)} + v^{i}(t)∇_{γ′(t)}X_{i}) ,

showing that the covariant derivative of the vector field V along γ is indeed well-defined, and does not depend on the choice of the vector fields X_{1}, X_{2}, …, X_{n}. The properties of the covariant derivative operator D/dt stated in the following lemma follow easily from the definition of the covariant derivative.

Lemma 3.5 Let M be a smooth manifold, let ∇ be an affine connection on M, and let γ: I → M be a smooth curve in M. Let V and W be smooth vector fields along γ and let f: I → ℝ be a smooth real-valued function. Then

(i) \frac{D(V (t) + W (t))}{dt} = \frac{DV (t)}{dt} + \frac{DW (t)}{dt},

(ii) \frac{D(f(t)V (t))}{dt} = \frac{df(t)}{dt}V (t) + f(t)\frac{DV (t)}{dt},

(iii) if V (t) = X_{γ(t)} for all t, where X is some smooth vector field defined

over an open set in M, then \frac{DV (t)}{dt} = ∇_{γ′(t)}X.

Moreover the differential operator D/dt is the unique operator on the space of smooth vector fields along the curve γ satisfying (i), (ii) and (iii).

A smooth vector field V along a smooth curve γ is said to be parallel if

\frac{DV (t)}{dt} = 0 for all t.

3.3 Vector Fields along Parameterized Surfaces

Let M be a smooth manifold, let U be a connected open set in ℝ^{m}, and let α: U → M be a smooth map from U to M. Given (t^{1}, t^{2}, …, t^{m}) ∈ U, we define

\frac{∂α(t^{1}, t^{2}, …, t^{m})}{∂t^{i}}

to be the velocity vector of the curve t ↦ α(t^{1}, …, t^{i−1}, t, t^{i+1}, …, t^{m}) at t = t^{i}. Then ∂α/∂t^{i} is a smooth vector field along the map α for i = 1, 2, …, m.

Let ∇ be an affine connection on M. Given any smooth vector field V along the map α, and given (t^{1}, t^{2}, …, t^{m}) ∈ U, we define

\frac{DV (t^{1}, t^{2}, …, t^{m})}{∂t^{i}}

to be the covariant derivative of the vector field

t ↦ V (t^{1}, …, t^{i−1}, t, t^{i+1}, …, t^{m})

along the curve t ↦ α(t^{1}, …, t^{i−1}, t, t^{i+1}, …, t^{m}) at t = t^{i}. Then the partial covariant derivative DV/∂t^{i} is a smooth vector field along the map α of i = 1, 2, …, m. We now consider the case m = 2.

Let M be a smooth manifold of dimension n. A smooth parameterized surface in M is a smooth map α: U → M defined on a connected open subset U on ℝ^{2}.

Lemma 3.6 Let M be a smooth manifold and let ∇ be an affine connection on M. Let V be a smooth vector field along a smooth parameterized surface α: U → M in M. Then

\frac{D}{∂s} \frac{∂α(s, t)}{∂t} − \frac{D}{∂t} \frac{∂α(s, t)}{∂s} = T (\frac{∂α(s, t)}{∂s}, \frac{∂α(s, t)}{∂t}),
\frac{D}{∂s} \frac{DV (s, t)}{∂t} − \frac{D}{∂t} \frac{DV (s, t)}{∂s} = R (\frac{∂α}{∂s}, \frac{∂α}{∂t}) V (s, t),

where T and R are the torsion and curvature tensors of the affine connection ∇.

Proof Without loss of generality, we may suppose that the image of the map α: U → M is contained in the domain of some smooth coordinate system (x^{1}, x^{2}, …, x^{n}). Let B_{1}, B_{2}, …, B_{n} be the smooth vector fields over this coordinate patch defined by B_{i} = ∂/∂x^{i} for i = 1, 2, …, n. The vectors B_{1}, B_{2}, …, B_{n} constitute a basis of the tangent space at each point of this coordinate patch. Moreover [B_{j}, B_{k}] = 0 for all j and k, so that

∇_{B_{j}}B_{k} − ∇_{B_{k}}B_{j} = T (B_{j}, B_{k}), ∇_{B_{j}}∇_{B_{k}}B_{i} − ∇_{B_{k}}∇_{B_{j}}B_{i} = R(B_{j}, B_{k})B_{i},

for all i, j and k.

The map α: U → M is specified, with respect to the coordinate system (x^{1}, x^{2}, …, x^{n}), by smooth real-valued functions α^{1}, α^{2}, …, α^{n} on U. (Thus the coordinates of α(s, t) are given by x^{i} = α^{i}(s, t) for all i = 1, 2, …, n.) It follows that

\frac{∂α}{∂s} = ∑_{j=1}^{n} \frac{∂α^{j}}{∂s} B_{j}, \frac{∂α}{∂t} = ∑_{k=1}^{n} \frac{∂α^{k}}{∂t} B_{k}.

Thus

\frac{DX}{∂s} = ∑_{j=1}^{n} \frac{∂α^{j}}{∂s} ∇_{B_{j}}X, \frac{DX}{∂t} = ∑_{k=1}^{n} \frac{∂α^{k}}{∂t} ∇_{B_{k}}X.

for all smooth vector fields X on the coordinate patch. Now

\frac{D}{∂s} \frac{∂α}{∂t} = ∑_{k=1}^{n} \frac{D}{∂s}(\frac{∂α^{k}}{∂t} B_{k})

= ∑_{k=1}^{n} \frac{∂^{2}α^{k}}{∂s∂t} B_{k} + ∑_{j,k=1}^{n} \frac{∂α^{j}}{∂s} \frac{∂α^{k}}{∂t} ∇_{B_{j}}B_{k}.

Thus

\frac{D}{∂s} \frac{∂α}{∂t} − \frac{D}{∂t} \frac{∂α}{∂s} = ∑_{j,k=1}^{n} \frac{∂α^{j}}{∂s} \frac{∂α^{k}}{∂t} (∇_{B_{j}}B_{k} − ∇_{B_{k}}B_{j})

= ∑_{j,k=1}^{n} \frac{∂α^{j}}{∂s} \frac{∂α^{k}}{∂t} T (B_{j}, B_{k}) = T (\frac{∂α}{∂s}, \frac{∂α}{∂t}).

Let f: U → ℝ be a smooth real-valued function on U, and let V be a smooth vector field along the map α. Then

\frac{D}{∂s} \frac{D(fV)}{∂t} = \frac{D}{∂s}(\frac{∂f}{∂t} V + f \frac{DV}{∂t} )
= \frac{∂^{2}f}{∂s∂t}V + \frac{∂f}{∂t} \frac{DV}{∂s} + \frac{∂f}{∂s} \frac{DV}{∂t} + f \frac{D}{∂s} \frac{DV}{∂t} ,

and thus

(\frac{D}{∂s} \frac{D}{∂t} − \frac{D}{∂t} \frac{D}{∂s})(fV) = f(\frac{D}{∂s} \frac{DV}{∂t} − \frac{D}{∂t} \frac{DV}{∂s} )

Now any smooth vector field V along the map α can be expressed in the form

V (s, t) = ∑_{i=1}^{n} v^{i}(s, t)(B_{i})_{α(s,t)}

for some smooth real-valued functions v^{1}, v^{2}, …, v^{n} on U. It follows that

\frac{D}{∂s} \frac{DV}{∂t} − \frac{D}{∂t} \frac{DV}{∂s} = ∑_{i=1}^{n} v^{i}(\frac{D}{∂s} \frac{D}{∂t} − \frac{D}{∂t} \frac{D}{∂s})B_{i}.

But

\frac{D}{∂s} \frac{D}{∂t}B_{i} = ∑_{k=1}^{n} \frac{D}{∂s}(\frac{∂α^{k}}{∂t} ∇_{B_{k}}B_{i})

= ∑_{k=1}^{n} \frac{∂^{2}α^{k}}{∂s∂t} ∇_{B_{k}}B_{i} + ∑_{j,k=1}^{n} \frac{∂α^{j}}{∂s} \frac{∂α^{k}}{∂t} ∇_{B_{j}}∇_{B_{k}}B_{i},

and hence

(\frac{D}{∂s} \frac{D}{∂t} − \frac{D}{∂t} \frac{D}{∂s})B_{i} = ∑_{j,k=1}^{n} \frac{∂α^{j}}{∂s} \frac{∂α^{k}}{∂t} (∇_{B_{j}}∇_{B_{k}}B_{i} − ∇_{B_{k}}∇_{B_{j}}B_{i})

= ∑_{j,k=1}^{n} \frac{∂α^{j}}{∂s} \frac{∂α^{k}}{∂t} R(B_{j}, B_{k})B_{i}

= R(\frac{∂α}{∂s}, \frac{∂α}{∂t})B_{i}.

We deduce that

\frac{D}{∂s} \frac{DV}{∂t} − \frac{D}{∂t} \frac{DV}{∂s} = ∑_{i=1}^{n} v^{i}R(\frac{∂α}{∂s}, \frac{∂α}{∂t})B_{i} = R(\frac{∂α}{∂s}, \frac{∂α}{∂t})V,

as required.

4 Riemannian Manifolds

Definition Let M be a smooth manifold. A Riemannian metric g on M assigns to any smooth vector fields X and Y on M a smooth function g(X, Y), where

g(X_{1}+X_{2}, Y) = g(X_{1}, Y)+g(X_{2}, Y), g(X, Y_{1}+Y_{2}) = g(X, Y_{1})+g(X, Y_{2}),

g(fX, Y) = f g(X, Y) = g(X, fY), g(X, Y) = g(Y, X)

for all smooth real-valued functions f and vector fields X, X_{1}, X_{2}, Y, Y_{1}, Y_{2}, and

g(X, X) > 0 wherever X ≠ 0.

Let (x^{1}, x^{2}, …, x^{n}) be a smooth coordinate system defined over some open set U in M. Define smooth real-valued functions g_{ij} on U by the formula

g_{ij} = g(\frac{∂}{∂x^{i}} , \frac{∂}{∂x^{j}} )

The n×n matrix (g_{ij}) is positive-definite at each point of U, since g(X, X) > 0 for all non-zero vector fields X on U. If X and Y are smooth vector fields on M then

X = ∑_{i=1}^{n} a^{i} \frac{∂}{∂x^{i}} , Y = ∑_{i=1}^{n} b^{i} \frac{∂}{∂x^{i}} ,

where a^{1}, a^{2}, …, a^{n} and b^{1}, b^{2}, …, b^{n} are smooth functions on U, and therefore

g(X, Y) = ∑_{i,j=1}^{n} g_{ij}a^{i}b^{j}.

We deduce that the value of g(X, Y) at any point m of M depends only on the values X_{m} and Y_{m} of X and Y at m.

Given tangent vectors X_{m} and Y_{m} at a point m of M, we denote by g(X_{m}, Y_{m}) the value of g(X, Y) at m, where X and Y are any smooth vector fields on M with values X_{m} and Y_{m} at m. Then (X_{m}, Y_{m}) ↦ g(X_{m}, Y_{m}) is a well-defined inner product on the tangent space T_{m}M to M at m. Thus one can regard a Riemannian metric on a smooth manifold M as a smooth assignment of an inner product to each tangent space of M.

Definition A Riemannian manifold (M, g) consists of a smooth manifold M together with a (smooth) Riemannian metric g on M.

Remark It should be noted that the metric tensor of the theory of general relativity is not a Riemannian metric, since it is not positive-definite at each point of the space-time manifold. It is in fact a ‘semi-Riemannian’ metric.

Lemma 4.1 Let M be a Riemannian manifold with Riemannian metric g. Let θ be a transformation mapping smooth vector fields on M to smooth functions on M. Suppose that

θ(X + Y) = θ(X) + θ(Y), θ(fX) = f θ(X)

for all smooth real-valued functions f and vector fields X and Y on M. Then there exists a unique smooth vector field V on M with the property that θ(X) = g(V, X) for all smooth vector fields X on M.

Proof First we verify the uniqueness of the vector field V. Let U be an open set in M and let V and W be vector fields on U with the property that g(V, X) = θ(X) = g(W, X) for all smooth vector fields X on U. Then g(V −W, X) = 0 on U for all vector fields X. In particular, g(V −W, V −W) = 0 on U. It follows from the definition of a Riemannian metric that V −W = 0, so that V = W on U. This proves the uniqueness of the vector field V.

Now suppose that the open set U is the domain of some smooth coordinate system (x^{1}, x^{2}, …, x^{n}). Let

g_{ij} = g(\frac{∂}{∂x^{i}} , \frac{∂}{∂x^{j}} ) (i, j = 1, 2, …, n)

Then (g_{ij}) is a matrix of smooth functions on U which is positive definite, and hence invertible, at each point of U. Let (g^{ij}) be the smooth functions on U characterized by the property that, at each point of U, the matrix (g^{ij}) is the inverse of (g_{ij}). Thus

∑_{j=1}^{n} g^{ij}g_{jk} = \{ \begin{matrix}1 & if i = k; \\ 0 & if i ≠ k. \end{matrix}

Define a smooth vector field V on U by

V = ∑_{i,j=1}^{n} θ_{i}g^{ij} \frac{∂}{∂x^{j}} ,

where

θ_{i} = θ(\frac{∂}{∂x^{i}} ) (i = 1, 2, …, n).

Let X be a smooth vector field on U, given by

X = ∑_{k=1}^{n} a^{k} \frac{∂}{∂x^{k}} .

Then

g(V, X) = ∑_{i,j,k=1}^{n} θ_{i}g^{ij}g_{jk}a^{k} = ∑_{i=1}^{n} θ_{i}a^{i} = θ(X)

on U. We thus obtain a smooth vector field V over any coordinate patch U with the property that g(V, X) = θ(X) for all vector fields X on U. If we are given two overlapping coordinate systems on M then the uniqueness result already proved shows that the vector fields over the coordinate patches obtained in the manner just described must agree on the overlap of the coordinate patches. Thus we obtain a smooth vector field V defined over the whole of M such that θ(X) = g(V, X) for all smooth vector fields X on M, as required.

4.1 The Levi-Civita Connection

Let (M, g) be a Riemannian manifold and let ∇ be an affine connection on M. We say that ∇ is compatible with the Riemannian metric g if

Z[g(X, Y)] = g(∇_{Z}X, Y) + g(X, ∇_{Z}Y)

for all smooth vector fields X, Y and Z on M. We shall show that on every Riemannian manifold there exists a unique torsion-free connection that is compatible with the Riemannian metric.

Lemma 4.2 Let ∇ be an affine connection on a Riemannian manifold M that is compatible with the Riemannian metric g. Let V and W be smooth vector fields along some smooth curve γ: I → M in M (where I denotes some open interval in ℝ). Then

\frac{d}{dt}g(V (t), W (t)) = g(\frac{DV (t)}{dt}, W (t)) + g(V (t), \frac{DW (t)}{dt})

(where DV/dt and DW/dt are the covariant derivatives of the vector fields V and W along the curve γ).

Proof Suppose that γ(t) ∈ U, where U is the domain of some smooth coordinate system (x^{1}, x^{2}, …, x^{n}) on M. Then

V (t) = ∑_{i=1}^{n} v^{i}(t)(B_{i})_{γ(t)}, W (t) = ∑_{j=1}^{n} w^{j}(t)(B_{j})_{γ(t)},

where v^{1}, v^{2}, …, v^{n} and w^{1}, w^{2}, …, w^{n} are smooth functions on γ^{−1}(U) and B_{i} = ∂/∂x^{i} for i = 1, 2, …, n. Thus

\frac{d}{dt}g(V (t), W (t)) = ∑_{i,j=1}^{n} \frac{d}{dt} (v^{i}(t)w^{j}(t)g(B_{i}, B_{j}))

= ∑_{i,j=1}^{n}(\frac{dv^{i}(t)}{dt}w^{j}(t)g(B_{i}, B_{j}) + v^{i}(t)\frac{dw^{j}(t)}{dt}g(B_{i}, B_{j})

+ v^{i}(t)w^{j}(t)(g(∇_{γ′(t)}B_{i}, B_{j}) + g(B_{i}, ∇_{γ′(t)}B_{j})))

= g(∑_{i=1}^{n}(\frac{dv^{i}(t)}{dt}B_{i} + v^{i}(t)∇_{γ′(t)}B_{i}), W (t))

+ g(V (t), ∑_{j=1}^{n}(\frac{dw^{j}(t)}{dt}B_{j} + w^{j}(t)∇_{γ′(t)}B_{j}))

= g(\frac{DV (t)}{dt}, W (t)) + g(V (t), \frac{DW (t)}{dt}),

as required.

Theorem 4.3 Let (M, g) be a Riemannian manifold. Then there exists a unique torsion-free affine connection ∇ on M compatible with the Riemannian metric g. This connection is characterized by the identity

2g(∇_{X}Y, Z) = X[g(Y, Z)] + Y [g(X, Z)] − Z[g(X, Y)]
+ g([X, Y], Z) − g([X, Z], Y) − g([Y, Z], X)

for all smooth vector fields X, Y and Z on M.

Proof Given smooth vector fields X, Y and Z on M, let A(X, Y, Z) be the smooth function on M defined by

A(X, Y, Z) = \frac{1}{2}(X[g(Y, Z)] + Y [g(X, Z)] − Z[g(X, Y)]
+ g([X, Y], Z) − g([X, Z], Y) − g([Y, Z], X)).

Then A(X, Y, Z_{1} + Z_{2}) = A(X, Y, Z_{1}) + A(X, Y, Z_{2}) for all smooth vector fields X, Y, Z_{1} and Z_{2} on M. Using the identities

[X, fZ] = f [X, Z] + X[f] Z, [Y, fZ] = f [Y, Z] + Y [f] Z

one can readily verify that A(X, Y, fZ) = f A(X, Y, Z) for all smooth realvalued functions f and vector fields X, Y and Z on M. On applying Lemma 4.1 to the transformation Z ↦ A(X, Y, Z), we see that there is a unique vector field ∇_{X}Y on M with the property that A(X, Y, Z) = g(∇_{X}Y, Z) for all smooth vector fields X, Y and Z on M. Moreover

∇_{X_{1}+X_{2}}Y = ∇_{X_{1}}Y + ∇_{X_{2}}Y, ∇_{X}(Y_{1} + Y_{2}) = ∇_{X}Y_{1} + ∇_{X}Y_{2}.

Straightforward calculations show that

g(∇_{fX}Y, Z) = A(fX, Y, Z) = f A(X, Y, Z) = g(f ∇_{X}Y, Z),
g(∇_{X}(fY), Z) = A(X, fY, Z) = f A(X, Y, Z) + X[f] g(Y, Z)
= g(f ∇_{X}Y + X[f] Y, Z),

for all smooth real-valued functions f on M, so that

∇_{fX}Y = f ∇_{X}Y, ∇_{X}(fY) = f ∇_{X}Y + X[f] Y.

These properties show that ∇ is indeed an affine connection on M. Moreover

A(X, Y, Z) − A(Y, X, Z) = g([X, Y], Z),

so that ∇_{X}Y − ∇_{Y} X = [X, Y]. Thus the affine connection ∇ is torsion-free. Also

g(∇_{X}Y, Z) + g(Y, ∇_{X}Z) = A(X, Y, Z) + A(X, Z, Y) = X[g(Y, Z)],

showing that the affine connection ∇ preserves the Riemannian metric.

Finally suppose that ∇′ is any torsion-free affine connection on M which preserves the Riemannian metric. Then

X[g(Y, Z)] = g(∇′_{X}Y, Z) + g(Y, ∇′_{X}Z),
Y [g(X, Z)] = g(∇′_{Y} X, Z) + g(X, ∇′_{Y} Z),
Z[g(X, Y)] = g(∇′_{Z}X, Y) + g(X, ∇′_{Z}Y).

A straightforward calculation (using the fact that ∇′ is torsion-free) shows that A(X, Y, Z) = g(∇′_{X}Y, Z). Therefore ∇′_{X}Y = ∇_{X}Y for all smooth vector fields X and Y on M, as required.

Let (M, g) be a Riemannian manifold. The unique torsion-free affine connection on M which preserves the Riemannian metric is known as the Levi-Civita connection on M.

Example Let M be a smooth n-dimensional submanifold of k-dimensional Euclidean space ℝ^{k}. Given (tangential) vector fields X and Y on M, we decompose the directional derivative ∂_{X}Y of Y along X as ∂_{X}Y = ∇_{X}Y + S(X, Y), where ∇_{X}Y is tangential to M and S(X, Y) is orthogonal to M. Then ∇ is a torsion-free affine connection on M (as was shown in an earlier example). Now the restriction to the tangent spaces of M of the standard inner product ⟨., .⟩ on ℝ^{k} gives a Riemannian metric g on M. Moreover

g(∇_{X}Y, Z) + g(Y, ∇_{X}Z) = ⟨∂_{X}Y, Z⟩ + ⟨Y, ∂_{X}Z⟩ = X[⟨Y, Z⟩] = X[g(Y, Z)]

for all vector fields X, Y and Z on M that are everywhere tangential to M. We conclude that the affine connection ∇ on M coincides with the Levi-Civita connection of the Riemannian manifold (M, g).

The Riemann curvature tensor R of a Riemannian manifold (M, g) is given by the formula

R(W, Z, X, Y) = g(W, R(X, Y)Z),

where

R(X, Y)Z = ∇_{X}∇_{Y} Z − ∇_{Y} ∇_{X}Z − ∇_{[X,Y]}Z

for all smooth vector fields X, Y and Z on M. It follows immediately from Lemma 3.3 that the value of R(W, Z, X, Y) at a point m of M depends only on the values W_{m}, Z_{m}, X_{m} and Y_{m} of the vector fields W, Z, X and Y at the point m.

Proposition 4.4 Let (M, g) be a Riemannian manifold. The Riemann curvature tensor on M satisfies the following identities:—

(i) R(W, Z, X, Y) = −R(W, Z, Y, X),

(ii) R(W, X, Y, Z) + R(W, Y, Z, X) + R(W, Z, X, Y) = 0,

(iii) R(W, Z, X, Y) = −R(Z, W, X, Y),

(iv) R(W, Z, X, Y) = R(X, Y, W, Z)

for all smooth vector fields X, Y, Z and W on M.

Proof (i) follows directly from the definition of the Riemann curvature tensor, and (ii) corresponds to the First Bianchi Identity

R(X, Y)Z + R(Y, Z)X + R(Z, X)Y = 0

(see Lemma 3.4). Now

X[Y [g(W, Z)]] = X [g(∇_{Y} W, Z) + g(W, ∇_{Y} Z)]
= g(∇_{X}∇_{Y} W, Z) + g(∇_{Y} W, ∇_{X}Z)
+ g(∇_{X}W, ∇_{Y} Z) + g(W, ∇_{X}∇_{Y} Z),

and hence

[X, Y][g(W, Z)] = X[Y [g(W, Z)]] − Y [X[g(W, Z)]]
= g(∇_{X}∇_{Y} W − ∇_{Y} ∇_{X}W, Z)
+ g(W, ∇_{X}∇_{Y} Z − ∇_{Y} ∇_{X}Z).

Therefore

R(W, Z, X, Y) + R(Z, W, X, Y)
= g(W, R(X, Y)Z) + g(R(X, Y)W, Z)
= [X, Y][g(W, Z)] − g(∇_{[X,Y]}W, Z) − g(W, ∇_{[X,Y]}Z)
= 0.

This proves (iii). Using (i), (ii) and (iii), we see that

2R(W, Z, X, Y) = R(W, Z, X, Y) − R(Z, W, X, Y)
= −R(W, X, Y, Z) − R(W, Y, Z, X)
+ R(Z, X, Y, W) + R(Z, Y, W, X)
= (R(X, W, Y, Z) + R(X, Z, W, Y))
+ (R(Y, W, Z, X) + R(Y, Z, X, W))
= −R(X, Y, Z, W) − R(Y, X, W, Z)
= 2R(X, Y, W, Z).

This proves (iv).

Let m be point in M and let P be a two-dimensional vector subspace (plane) in the tangent space T_{m}M to M at m. Let (E_{1}, E_{2}) be an orthonormal basis of P. We define the sectional curvature K(P) of M in the plane P by the formula

K(P) = R(E_{1}, E_{2}, E_{1}, E_{2}).

Note that if X and Y are tangent vectors in P then

X = a_{11}E_{1} + a_{12}E_{2}, Y = a_{21}E_{1} + a_{22}E_{2},

for some real numbers a_{11}, a_{12}, a_{21} and a_{22}, and hence

R(X, Y, X, Y) = R(X, Y, a_{11}E_{1} + a_{12}E_{2}, a_{21}E_{1} + a_{22}E_{2})
= (a_{11}a_{22} − a_{12}a_{21})R(X, Y, E_{1}, E_{2})
= (det A)R(X, Y, E_{1}, E_{2}) = (det A)^{2}R(E_{1}, E_{2}, E_{1}, E_{2})
= (det A)^{2}K(P),

where A is the matrix given by

A = ( \begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix} ) .

In particular, if (X, Y) is any orthonormal basis of P then the matrix A is an orthogonal matrix, and thus det A = ±1. It follows that the value of the sectional curvature K(P) does not depend on the choice of the orthonormal basis (E_{1}, E_{2}) of P.

Lemma 4.5 Let (M, g) be a Riemannian manifold, and let m be a point of M. Then the values of the sectional curvatures K(P) for all planes P in the tangent space T_{m}M to M at m determine the Riemann curvature tensor at m.

Proof The calculation given above shows that the sectional curvatures determine the values of R(X, Y, X, Y) for all X, Y ∈ T_{m}M.

Now suppose that we are given X, Y, Z ∈ T_{m}M. Using the symmetries of the Riemann curvature tensor listed in Proposition 4.4, we see that

2R(X, Y, X, Z) = R(X, Y, X, Z) + R(X, Z, X, Y)
= R(X, Y + Z, X, Y + Z) − R(X, Y, X, Y)
− R(X, Z, X, Z).

Thus the sectional curvatures K(P) determine the values of R(X, Y, X, Z) for all tangent vectors X, Y and Z at m. It follows from this that the sectional curvatures determine R(X, Y, Z, X), R(Y, X, X, Z) and R(Y, X, Z, X). But

3R(W, X, Y, Z) = 2R(W, X, Y, Z) − R(W, Y, Z, X) − R(W, Z, X, Y)
= (R(W, X, Y, Z) + R(W, Y, X, Z))
+ (R(W, X, Y, Z) + R(W, Z, Y, X))
= R(W, X + Y, X + Y, Z) − R(W, X, X, Z)
− R(W, Y, Y, Z) + R(W, X + Z, Y, X + Z)
− R(W, X, Y, X) − R(W, Z, Y, Z).

We conclude that R(W, X, Y, Z) is determined by the sectional curvatures of M, as required.

5 Geometry of Surfaces in ℝ^{3}

A smooth surface in ℝ^{3} is defined to be a smooth 2-dimensional submanifold of ℝ^{3}. Let M be a smooth surface in ℝ^{3}. If this surface is orientable, then we can choose a unit normal vector ν_{p} at each point p of M (where ν_{p} is orthogonal to the tangent space T_{p}M to M at p) such that ν_{p} varies smoothly with p. We shall assume that M is orientable and that such a smooth unit normal vector field ν has been chosen on M.

Recall that if X and Y are vector fields on M then we denote by ∂_{X_{p}}Y the derivative of the vector field Y along X. Thus if X_{p} is the tangent vector to the smooth curve γ: (−ε, ε) → M at zero then

∂_{X_{p}}Y ≡ \frac{dY (γ(t))}{dt}|_{t=0} .

Note that this definition makes sense for any vector field Y along the surface M which takes values in ℝ^{3}, whether or not that vector field is everywhere tangential to M. In particular, we can define ∂_{X}ν, where ν is the chosen unit normal vector field on M by

∂_{X_{p}}ν ≡ \frac{dν(γ(t))}{dt}|_{t=0} ,

where γ is a smooth curve such that γ(0) = p and γ′(0) = X_{p}.

Now let X and Y be two vector fields on M which are everywhere tangential to the surface M. We split ∂_{X}Y into components tangential to the surface and normal to the surface. We denote the tangential component by ∇_{X}Y Thus

∂_{X}Y = ∇_{X}Y + σ(X, Y)ν

where σ(X, Y) is a smooth function on M determined by the vector fields X and Y. Now

σ(X + Y, Z) = σ(X, Z) + σ(Y, Z), σ(X, Y + Z) = σ(X, Y) + σ(X, Z).

Next we note that

σ(fX, Y) = fσ(X, Y) = σ(X, fY)

since

∂_{fX}Y = f∂_{X}Y, ∂_{X}(fY) = f∂_{X}Y + X[f]Y,

and X[f]Y is clearly tangential to M. Also σ(X, Y) = σ(Y, X) for all tangential vector fields X and Y on M, since

∂_{X}Y − ∂_{Y} X = [X, Y],

and the Lie bracket [X, Y] is tangential to M.

We can regard the surface M as a 2-dimensional Riemannian manifold, where the Riemannian metric on M is the restriction to each tangent space of the standard inner product ⟨., .⟩ on ℝ^{3}. Then the connection ∇ on M defined as above is torsion-free and preserves the Riemannian metric (i.e.,

Z[⟨X, Y ⟩] = ⟨∇_{Z}X, Y ⟩ + ⟨X, ∇_{Z}Y ⟩

for all tangential vector fields X, Y and Z on M. Thus this connection is the Levi-Civita connection on the surface M.

Observe that ∂_{X}ν is tangential to X for all tangental vectors X on M (where ∂_{X}ν denotes the directional derivative of the unit normal vector field ν along X. This follows from the fact that

2 ⟨∂_{X}ν, ν⟩ = X[|ν|^{2}] = 0.

Let Y be a vector field on M that is everywhere tangential to M. Then ⟨ν, Y ⟩ = 0, hence

⟨∂_{X}ν, Y ⟩ = ∂_{X} ⟨ν, Y ⟩ − ⟨ν, ∂_{X}Y ⟩
= −σ(X, Y).

Observe that if X and Y are vector fields on M everywhere tangential to M and if V is a vector field on M taking values in ℝ^{3} (i.e., V is a ‘vector field along the embedding map i: M ↪ ℝ^{3}’) then

∂_{X}∂_{Y} V − ∂_{Y} ∂_{X}V = ∂_{[X,Y]}V.

Indeed if V = (v^{1}, v^{2}, v^{3}) then

∂_{X}V ≡ (X[v^{1}], X[v^{2}], X[v^{3}]), ∂_{Y} V ≡ (Y [v^{1}], Y [v^{2}], Y [v^{3}]),

so that

∂_{X}∂_{Y} V − ∂_{Y} ∂_{X}V

= (X[Y [v^{1}]] − Y [X[v^{1}]], X[Y [v^{2}]] − Y [X[v^{2}]], X[Y [v^{3}]] − Y [X[v^{3}]])
= ([X, Y][v^{1}], [X, Y][v^{2}], [X, Y][v^{3}]) = ∂_{[X,Y]}V.

Let W, Z, X and Y be smooth vector fields on M, everywhere tangential to M. Then

⟨W, ∂_{X}∂_{Y} Z⟩ = ⟨W, ∂_{X}(∇_{Y} Z + σ(Y, Z)ν)⟩
= ⟨W, ∂_{X}∇_{Y} Z⟩ + σ(Y, Z) ⟨W, ∂_{X}ν⟩
= ⟨W, ∂_{X}∇_{Y} Z⟩ + σ(Y, Z) ⟨W, ∂_{X}ν⟩
= ⟨W, ∇_{X}∇_{Y} Z⟩ − σ(Y, Z)σ(X, W).

Using the fact that

∂_{X}∂_{Y} Z − ∂_{Y} ∂_{X}Z = ∂_{[X,Y]}Z.

we see that the Riemann curvature tensor R of M is given by

R(W, Z, X, Y) ≡ ⟨W, ∇_{X}∇_{Y} Z − ∇_{Y} ∇_{X}Z − ∇_{[X,Y]}Z⟩
= ⟨W, ∂_{X}∂_{Y} Z − ∂_{Y} ∂_{X}Z⟩
+ σ(Y, Z)σ(X, W) − σ(X, Z)σ(Y, W)
− ⟨W, ∂_{[X,Y]}Z⟩
= σ(Y, Z)σ(X, W) − σ(X, Z)σ(Y, W).

This identity expresses the Riemann curvature tensor of the surface M in terms of the second fundamental form of the surface. This result is essentially due to Gauss (though Gauss employed an alternative formalism).

We now derive an identity satisfied by the second fundamental form σ on M. Note that

⟨ν, ∂_{X}∂_{Y} Z⟩ = ⟨ν, ∂_{X}(∇_{Y} Z + σ(Y, Z)ν)⟩
= σ(X, ∇_{Y} Z) + ⟨ν, X[σ(Y, Z)]ν + σ(Y, Z)∂_{X}ν⟩ .

But ⟨ν, ∂_{X}ν⟩ = 0, hence

⟨ν, ∂_{X}∂_{Y} Z⟩ = σ(X, ∇_{Y} Z) + X[σ(Y, Z)].

Also

⟨ν, ∂_{[X,Y]}Z⟩ = σ([X, Y], Z) = σ(∇_{X}Y, Z) − σ(∇_{Y} X, Z).

But

⟨ν, ∂_{X}∂_{Y} Z⟩ − ⟨ν, ∂_{X}∂_{Y} Z⟩ = ⟨ν, ∂_{[X,Y]}Z⟩ .

Therefore

X[σ(Y, Z)] − Y [σ(X, Z)] + σ(X, ∇_{Y} Z) − σ(Y, ∇_{X}Z)

= σ(∇_{X}Y, Z) − σ(∇_{Y} X, Z),

so that

(∇_{X}σ)(Y, Z) = (∇_{Y} σ)(X, Z),

where ∇σ denotese the covariant derivative of the tensor σ, defined by

(∇_{X}σ)(Y, Z) ≡ X[σ(Y, Z)] − σ(∇_{X}Y, Z) − σ(Y, ∇_{X}Z).

Gathering together the identities proved above, we see that we have verified the Gauss-Codazzi identities, stated in the following lemma.

Lemma 5.1 (The Gauss-Codazzi Identities) Let M be a smooth surface embedded in ℝ^{3}. Then the Riemann curvature tensor R on M is expressed in terms of the second fundamental form σ of the embedding of M in ℝ^{3} by the identity

R(W, Z, X, Y) = σ(W, X)σ(Z, Y) − σ(W, Y)σ(Z, X). (Gauss’ Identity)

Also the covariant derivative ∇σ of the second fundamental form σ satisfies the identity

(∇_{X}σ)(Y, Z) = (∇_{Y} σ)(X, Z). (Codazzi’s Identity)

Let (E_{1}, E_{2}) be an orthonormal basis of the tangent space T_{p}M to M at the point p. The Gaussian curvature of M at p is defined to be the quantity K(p) defined by

K(p) = R(E_{1}, E_{2}, E_{1}, E_{2}).

The Gaussian curvature K(p) of M at p is well-defined independently of the choice of orthonormal basis (E_{1}, E_{2}) of the tangent space T_{p}M to M at p. It follows from the Gauss-Codazzi equations that

K(p) = σ(E_{1}, E_{1})σ(E_{2}, E_{2}) − σ(E_{1}, E_{2})σ(E_{2}, E_{1}).

Thus if (s_{ij}) is the 2 × 2 matrix representing σ at p with respect to the orthonormal basis (E_{1}, E_{2}), then K(p) is the determinant of this matrix.

Lemma 5.2 Let M be a smooth surface in ℝ^{3} and let (X, Y) be a smooth moving frame on M. Then the Gaussian curvature K is given by

K = \frac{σ(X, X)σ(Y, Y) − σ(X, Y)^{2}}{⟨X, X⟩ ⟨Y, Y ⟩ − ⟨X, Y ⟩^{2}}

Proof Let (E_{1}, E_{2}) be an orthonormal moving frame defined around some point of M. We can write

X = a_{11}E_{1} + a_{12}E_{2}
Y = a_{21}E_{1} + a_{22}E_{2}.

Let us define matrices A, B, and C by

A = ( \begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix} ) ,
B = ( \begin{matrix}σ(E_{1}, E_{1}) & σ(E_{1}, E_{2}) \\ σ(E_{2}, E_{1}) & σ(E_{2}, E_{2}) \end{matrix} ) .
C = ( \begin{matrix}σ(X, X) & σ(X, Y) \\ σ(Y, X) & σ(Y, Y) \end{matrix} ) .

The matrices B and C are symmetric and C = A^{T} BA, where A^{T} is the transpose of A. Moreover the Gaussian curvature K of M is the determinant det B of the matrix B. Thus det C = K(det A)^{2}. But

A^{T} A = ( \begin{matrix}⟨X, X⟩ & ⟨X, Y ⟩ \\ ⟨Y, X⟩ & ⟨Y, Y ⟩ \end{matrix} ) ,

so that

(det A)^{2} = ⟨X, X⟩ ⟨Y, Y ⟩ − ⟨X, Y ⟩^{2} .

The required result follows immediately from this.

Let S^{2} denote the standard unit sphere in ℝ^{3}. We can regard the unit normal vector field ν on the smooth surface M as defining a smooth map from M to S^{2} (since ν(p) is a vector of unit length in ℝ^{2}). This smooth map from M to S^{2} is customarily referred to as the Gauss map.

Let us consider the derivative ν_{∗}: T_{p}M → T_{ν(p)}S^{2} of the Gauss map ν: M → S^{2} at p. If γ: (−ε, ε) → M is a smooth curve in M with γ(0) = p and γ′(0) = X_{p} then

ν_{∗}X_{p} = \frac{dν(γ(t))}{dt}|_{t=0} = ∂_{X_{p}}ν.

Note that the tangent space T_{ν(p)}S^{2} of S^{2} at ν(p) is parallel to the tangent space T_{p}M of M at p (since both are orthogonal to the vector ν(p)). Using the fact that

⟨∂_{X_{p}}ν, Y_{p}⟩ = −σ(X_{p}, Y_{p})

we see that if (E_{1}, E_{2}) is an orthonormal basis of the tangent space to M at p and if (Ẽ_{1}, Ẽ_{2}) is the corresponding orthonormal basis of T_{ν(p)}S^{2} (where the components of E_{1} and E_{2} with respect to the standard basis of ℝ^{3} are equal to the corresponding components of Ẽ_{1} and Ẽ_{2} respectively) then

ν_{∗}E_{1} = −σ(E_{1}, E_{1})Ẽ_{1} − σ(E_{2}, E_{1})Ẽ_{2}
ν_{∗}E_{2} = −σ(E_{1}, E_{2})Ẽ_{1} − σ(E_{2}, E_{2})Ẽ_{2}

Thus the matrix representing the derivative of the Gauss map ν: M → S^{2} at p with respect to these orthonormal bases is the symmetric matrix

( \begin{matrix}−s_{11} & −s_{12} \\ −s_{21} & −s_{22} \end{matrix} ) ,

where s_{ij} = σ(E_{1}, E_{2}). Now the Gaussian curvature K(p) at p is the determinant of this matrix (see above). Therefore the derivative ν_{∗}: T_{p}M → T_{ν(p)}S^{2} of the Gauss map ν: M → S^{2} at p multiplies areas by a factor of |K(p)|. Using this fact we deduce the following result.

Lemma 5.3 Let M be a smooth oriented surface in ℝ^{3} and let ν: M → S^{2} be the Gauss map of this surface. Suppose that ν is injective on a region D of M. Then

Area of ν(D) = ∫_{D} |K|dA,

where the integral is taken with respect to the usual area measure on A.

Informally we can regard the magnitude |K| of the Gaussian curvature at p as the limit of the ratio

\frac{area of ν(D)}{area of D}

as the region D in M shrinks down inside ever smaller neighbourhoods of the point p. If K(p) > 0 then the Gauss map ν: M → S^{2} is orientation-preserving around p. If K(p) < 0 then the Gauss map is orientation-reversing around p.

However although this description of the Gaussian curvature depends on the manner in which the surface M is embedded in ℝ^{3}, yet the expression for the Gaussian curvature in terms of the Riemann curvature tensor of M (with respect to the metric on M induced from the standard inner product on ℝ^{3}) shows that the Gaussian curvature of M is completely determined by the induced Riemannian metric on M. Thus isometric surfaces in ℝ^{3} have the same Gaussian curvature at corresponding points of the surfaces.

(Recall that an isometry φ: M_{1} → M_{2} is a diffeomorphism which preserves the lengths of all vectors.) The well-known mathematician Gauss discovered this fact, expressing it in his Theorema Egregium (which means ‘remarkable theorem’ in Latin).

Example Let us consider a surface M in ℝ^{3} of the form z = f(x, y), where f is a smooth function of x and y. Let us write

f_{x} ≡ \frac{∂f}{∂x}, f_{y} ≡ \frac{∂f}{∂y}

f_{xx} ≡ \frac{∂^{2}f}{∂x^{2}} ,  f_{yy} ≡ \frac{∂^{2}f}{∂y^{2}} ,  f_{xy} = f_{yx} ≡ \frac{∂^{2}f}{∂x∂y} .

We let X and Y be the smooth vector fields tangent to M given by

X = (1, 0, f_{x}(x, y)), Y = (0, 1, f_{y}(x, y)).

Note that (X, Y) is a smooth moving frame on M. We let ν be the unit normal vector field on M given by

ν = \frac{1}{\sqrt{1 + (f_{x})^{2} + (f_{y})^{2}}} (−f_{x}, −f_{y}, 1) .

Now

∂_{X}X = (0, 0, f_{xx}) ,
∂_{X}Y = ∂_{Y} X
= (0, 0, f_{xy}) ,
∂_{Y} Y = (0, 0, f_{yy}) ,

therefore the second fundamental form σ is given by

σ(X, X) = ⟨ν, ∂_{X}X⟩
= \frac{f_{xx}}{\sqrt{1 + (f_{x})^{2} + (f_{y})^{2}}} ,
σ(X, Y) = ⟨ν, ∂_{X}Y ⟩
= \frac{f_{xy}}{\sqrt{1 + (f_{x})^{2} + (f_{y})^{2}}} ,
σ(Y, X) = σ(X, Y),
σ(Y, Y) = ⟨ν, ∂_{X}Y ⟩
= \frac{f_{yy}}{\sqrt{1 + (f_{x})^{2} + (f_{y})^{2}}} .

Now

⟨X, X⟩ ⟨Y, Y ⟩ − ⟨X, Y ⟩^{2} = (1 + (f_{x})^{2}) (1 + (f_{y})^{2}) − (f_{x})^{2}(f_{y})^{2}
= 1 + (f_{x})^{2} + (f_{y})^{2}.

The Gaussian curvature of the surface M is therefore given by

K = \frac{f_{xx}f_{yy} − (f_{xy})^{2}}{(1 + (f_{x})^{2} + (f_{y})^{2})^{2}} .

Now let M be an oriented smooth surface in ℝ^{3} and let p be a point of M. By performing a translation we can ensure that the point p is at the origin (0, 0, 0) of ℝ^{3}. Moreover, by performing a rotation of the surface, we can ensure that the tangent space to M at p is given by the plane z = 0.

An application of the Inverse Function Theorem shows that the surface M can be represented around p by an equation of the form z = f(x, y). Indeed if π: M → ℝ^{2} is the map sending (x, y, z) ∈ M to (x, y) then the derivative of π at p is an isomorphism. Therefore π has a continuous inverse π^{−1}: U → M which maps some open neighbourhood U of (0, 0) in ℝ^{2} onto an open neighbourhood of p in M. If φ: M → ℝ is the restriction to M of the function that sends (x, y, z) to z then M is given by the equation z = f(x, y) around p, where f = φ ∘ π^{−1}.

Using the fact that the tangent space to M at p is the plane z = 0 (where p is the origin of ℝ^{3}), we see that f_{x}(0, 0) = 0 and f_{y}(0, 0) = 0. It follows from the formula for the Gaussian curvature calculated in the example above that

K(p) = f_{xx}f_{yy} − (f_{xy})^{2},

where

f_{xx} ≡ \frac{∂^{2}f}{∂x^{2}} ,  f_{yy} ≡ \frac{∂^{2}f}{∂y^{2}} ,  f_{xy} = f_{yx} ≡ \frac{∂^{2}f}{∂x∂y} .

Now Taylor’s theorem implies that

f(x, y) = \frac{1}{2} (f_{xx}x^{2} + 2f_{xy}xy + f_{yy}y^{2}) + O ((x^{2} + y^{2})^{\frac{3}{2}} )

(where O ((x^{2} + y^{2})^{\frac{3}{2}} ) is a quantity with the property that

(x^{2} + y^{2})^{− \frac{3}{2}} O ((x^{2} + y^{2})^{\frac{3}{2}} )

remains bounded as (x, y) → (0, 0)). Thus the paraboloid represented by the equation

z = \frac{1}{2} (f_{xx}x^{2} + 2f_{xy}xy + f_{yy}y^{2})

is the paraboloid which best approximates to the surface M around the point p. This paraboloid is known as the osculating paraboloid at the point p of M. We can find such an osculating paraboloid approximating the surface M around each point of M. The osculating paraboloid at any point of M is completely determined by the second fundamental form of M at that point. Indeed if p is a point of M then the osculating paraboloid at p is the paraboloid consisting of all points of the form

\{q ∈ ℝ^{3} : q = p + X_{p} + \frac{1}{2}σ(X_{p}, X_{p})ν_{p} for some X_{p} ∈ T_{p}M\}.

It follows from the theorem concerning diagonalization of symmetric matrices that, for each point p of M there exists an orthonormal basis (E_{1}, E_{2}) of the tangent space T_{p}M to M at p such that σ(E_{1}, E_{2}) = 0. Let us denote σ(E_{1}, E_{1}) and σ(E_{2}, E_{2}) by κ_{1} and κ_{2} respectively. The quantities κ_{1} and κ_{2} are known as the principal curvatures of M at p. The lines represented by the vectors E_{1} and E_{2} are referred to as the principal directions at p.

Now the Gaussian curvature K of M at p is equal to the determinant of the matrix

( \begin{matrix}σ(E_{1}, E_{1}) & σ(E_{1}, E_{2}) \\ σ(E_{2}, E_{1}) & σ(E_{2}, E_{2}) \end{matrix} )

and is therefore equal to the product κ_{1}κ_{2} of the principal curvatures of M at p. The mean curvature H at p is defined by

H(p) ≡ \frac{1}{2} (σ(E_{1}, E_{1}) + σ(E_{2}, E_{2}))

(where (E_{1}, E_{2}) is an orthonormal basis of the tangent space T_{p}M to M at p). If κ_{1} and κ_{2} are the principal curvatures at p then

H = \frac{1}{2}(κ_{1} + κ_{2}).

Unlike the Gaussian curvature K, the mean curvature H does not depend simply on the induced Riemannian metric on M but depends on the manner in which the surface M is embedded in ℝ^{3}. It is not invariant under isometries (unlike the Gaussian curvature). The principal curvatures κ_{1} and κ_{2} at the point p are given by the formula H ± \sqrt{H^{2} − K} (where K is the Gaussian curvature and H is the mean curvature at p).

The shape of the surface M around the point p approximates to the shape of the osculating paraboloid at p, which is in turn determined by the principal curvatures κ_{1} and κ_{2}.

Consider the case when κ_{1} and κ_{2} are both non-zero and have the same sign, so that the second fundamental form is either positive-definite or negative-definite. This is the case if and only if K > 0. In this case the osculating paraboloid to the surface M at p is bowl-shaped, and it is easy to prove that all points of M sufficiently close to p all lie on one side of the tangent plane to M at p. Such a point p of M at which the Gaussian curvature K satisfies K > 0 is referred to as an elliptic point of M.

Next consider the case when κ_{1} and κ_{2} are non-zero and have opposite signs, so that the second fundamental form is non-degenerate and indefinite. This is the case if and only if K < 0. In this case the osculating paraboloid is saddle-shaped, and the surface M around p lies on both sides of the tangent plane to M at p. Such a point at which the Gaussian curvature K satisfies K < 0 is referred to as a hyperbolic point of M.

Next consider the case when exactly one of the principal curvatures κ_{1} and κ_{2} is non-zero, so that the second fundamental form is degenerate but is non-zero. This is the case if and only if K = 0 and H ≠ 0. In this case the osculating paraboloid is trough-shaped. Such a point at which K = 0 and H ≠ 0 is referred to to as a parabolic point of M.

The remaining case occurs when both κ_{1} and κ_{2} are zero, so that the second fundamental form vanishes. This is the case if and only if K = 0 and H = 0. The osculating paraboloid at p is then the tangent plane at p. Such a point is referred to as a planar point of M.

A point p of M is said to be an umbilic point if and only if κ_{1} = κ_{2}. This is the case if and only if H^{2} = K. An umbilic point is either an ellipic point or a planar point.

Example Let M be a sphere of radius R about the origin in ℝ^{3}. The Gauss map sends a point (x, y, z) of M to the point

(\frac{x}{R}, \frac{y}{R}, \frac{z}{R})

on the unit sphere S^{2}. Therefore the derivative of the Gauss map ν: M → S^{2} sends a vector (a^{1}, a^{2}, a^{3}) tangent to M at a point (x, y, z) of M to the vector

(\frac{a^{1}}{R} , \frac{a^{2}}{R} , \frac{a^{3}}{R} )

We conclude that

σ(X, Y) = −\frac{1}{R} ⟨X, Y ⟩

(with respect to the outward normal direction) for all tangent vectors X and Y on M. The principal curvatures are equal to −R^{−1} everywhere on M. The mean curvature is everywhere equal to −R^{−1} and the Gaussian curvature is everywhere equal to R^{−2}. Every point on the sphere M of radius R is an elliptic point and an umbilic point.

Let M be an oriented smooth surface in ℝ^{3} and let ν be the normal vector field on M. Let γ: I_{γ} → M be a curve in M parameterized by arclength such that γ(0) = p. The normal curvature of γ at γ(s) is defined to be the quantity

⟨ν_{γ(s)}, \frac{d^{2}γ(s)}{ds^{2}}⟩

(where γ is parameterized by arclength). We claim that the normal curvature of γ at γ(s) is equal to σ(γ′(s), γ′(s)), where σ is the second fundamental form. Indeed let us choose an orthonormal moving frame (E_{1}, E_{2}) around γ(s). Then we can write

\frac{dγ(s)}{dt} = a^{1}(s)E_{1} + a^{2}(s)E_{2}.

Therefore

\frac{d^{2}γ(s)}{dt^{2}} = \frac{da^{1}(s)}{dt}E_{1} + \frac{da^{2}(s)}{dt}E_{2} + a^{1}(s)∂_{γ′(s)}E_{1} + a^{2}(s)∂_{γ′(s)}E_{2},

so that

⟨ν_{γ(s)}, \frac{d^{2}γ(s)}{ds^{2}}⟩ = a^{1}(s)σ(γ′(s), E_{1}) + a^{2}(s)σ(γ′(s), E_{2})
= σ(γ′(s), γ′(s)).

as required. Now γ′(t) is a vector of unit length (since γ is parameterized by arclength). Therefore if κ_{1} and κ_{2} are the principal curvatures of M at γ(s), where κ_{1} ≤ κ_{2}, then the value of the normal curvature of γ at γ(s) lies between κ_{1} and κ_{2}. More precisely there is an orthonormal basis (E_{1}, E_{2}) of the tangent space at γ(s) such that κ_{1} = σ(E_{1}, E_{1}), κ_{2} = σ(E_{2}, E_{2}) and σ(E_{1}, E_{2}) = 0 (i.e., the vectors E_{1} and E_{2} point along the principal directions at γ(s)). Let us write

γ′(s) = cos φ E_{1} + sin φ E_{2}

for some angle φ. Then the normal curvature of γ at γ(s) is

cos^{2} φ κ_{1} + sin^{2} φ κ_{2}.

6 Geodesics in Riemannian Manifolds

Let (M, g) be a Riemannian manifold of dimension n, and let γ: I → M be a smooth curve in M, defined over some interval I in ℝ. We say that γ is a geodesic if and only if

\frac{D}{dt} (\frac{dγ(t)}{dt}) = 0.

Thus γ is a geodesic if and only if the velocity vector field t ↦ γ′(t) is parallel along γ (with respect to the Levi-Civita connection on M). The geodesic γ: I → M is said to be maximal if it cannot be extended to a geodesic defined over some interval J, where I ⊂ J and I ≠ J.

Let γ: I → M be a geodesic in the Riemannian manifold (M, g). Then the length |γ′(t)| of the velocity vector γ′(t) is constant along the curve, since

\frac{d}{dt}|γ′(t)|^{2} = \frac{d}{dt}g (\frac{dγ(t)}{dt}, \frac{dγ(t)}{dt}) = g (\frac{D}{dt} \frac{dγ}{dt} , \frac{dγ}{dt} ) + g (\frac{dγ}{dt} , \frac{D}{dt} \frac{dγ}{dt} ) = 0.

Let us choose a smooth coordinate system (x^{1}, x^{2}, …, x^{n}) over some open set U in the smooth manifold M. Let the smooth functions Γ_{jk}^{i} on U be characterized by the property that

∇\frac{∂}{∂x^{j}} \frac{∂}{∂x^{k}} = ∑_{i=1}^{n} Γ_{jk}^{i} \frac{∂}{∂x^{i}} .

Then Γ_{jk}^{i} = Γ_{kj}^{i} for all j and k, since

∇\frac{∂}{∂x^{j}} \frac{∂}{∂x^{k}} − ∇\frac{∂}{∂x^{k}} \frac{∂}{∂x^{j}} = [\frac{∂}{∂x^{j}} , \frac{∂}{∂x^{k}} ] = 0.

Let γ: I → U be a smooth curve in U, and let γ^{i}(t) = x^{i} ∘ γ(t) for all t ∈ γ^{−1}(U). Then

\frac{dγ(t)}{dt} = ∑_{k=1}^{n} \frac{dγ^{k}(t)}{dt} \frac{∂}{∂x^{k}} ,

so that

\frac{D}{dt} \frac{dγ(t)}{dt} = ∑_{k=1}^{n} (\frac{d^{2}γ^{k}(t)}{dt^{2}} \frac{∂}{∂x^{k}} + \frac{dγ^{k}(t)}{dt} ∑_{j=1}^{n} \frac{dγ^{j}(t)}{dt}∇\frac{∂}{∂x^{j}} \frac{∂}{∂x^{k}} )

= ∑_{i=1}^{n} (\frac{d^{2}γ^{i}(t)}{dt^{2}} + ∑_{j=1}^{n} ∑_{k=1}^{n} Γ_{jk}^{i}(γ(t))\frac{dγ^{j}(t)}{dt} \frac{dγ^{k}(t)}{dt}) \frac{∂}{∂x^{i}} .

Thus γ: I → U is a geodesic if and only if

\frac{d^{2}γ^{i}(t)}{dt^{2}} + ∑_{j=1}^{n} ∑_{k=1}^{n} Γ_{jk}^{i}(γ(t))\frac{dγ^{j}(t)}{dt} \frac{dγ^{k}(t)}{dt} = 0 (i = 1, 2, …, n).

Standard existence and uniqueness theorems for solutions of ordinary differential systems of equations ensure that, given a tangent vector V at any point m of M, and given any real number t_{0}, there exists a unique maximal geodesic γ: I → M, defined on some open interval I containing t_{0}, such that γ(t_{0}) = m and γ′(t_{0}) = V. Moreover, for each point m of M, there exists some δ_{m} > 0 such that every tangent vector V at m whose length is less than δ satisfies V = γ′_{V} (0) for some geodesic γ_{V}, defined over some interval containing [0, 1]. The exponential map

exp_{m}: {V ∈ T_{m}M : |V | < δ_{m}} → M

at m is defined by exp_{m}(V) = γ_{V} (1), where γ_{V}: [0, 1] → M is the unique geodesic in M satisfying γ_{V} (0) = m and γ′_{V} (0) = V. Standard smoothness results for solutions of ordinary differential systems ensure that exp_{m}(V) depends smoothly on both V and m.

A diffeomorphism is a homeomorphism between smooth manifolds which is smooth and has a smooth inverse.

Lemma 6.1 Let (M, g) be a Riemannian manifold, let m be a point of M and let exp_{m}: D → M be the exponential map at p, defined over a neighbourhood D of zero in the tangent space T_{m}M at m. If D is chosen sufficiently small then exp_{m} maps D diffeomorphically onto some open set in m. The inverse of exp_{m} |D therefore provides a smooth chart around the point m.

Proof The tangent space T_{0}(T_{m}M) to T_{m}M at the zero vector is naturally isomorphic to T_{m}M itself. Moreover, given any V ∈ T_{m}M, the element of T_{0}(T_{m}M) that is mapped to V under this natural isomorphism is the velocity vector of the curve t ↦ tV at t = 0. But the exponential map exp_{m} maps the curve t ↦ tV in T_{m}M onto the geodesic γ_{V} in M with initial condition γ′_{V} (0) = V. We deduce that the derivative of exp_{m}: D → M at the zero vector sends the vector in T_{0}(T_{m}M) corresponding to V onto the vector V itself. We deduce that the derivative of the exponential map exp_{m} at the zero vector is an isomorphism of tangent spaces. It follows from the Inverse Function Theorem that the exponential map exp_{m} maps some sufficiently small open neighbourhood D of zero in T_{m}M diffeomorphically onto an open set in M, as required.

Lemma 6.2 Let γ: [a, b] → M be a smooth curve in a Riemannian manifold (M, g), and let X be a smooth vector field along the curve γ. Then there exists a smooth map α: [a, b] × (−ε, ε) → M such that α(t, 0) = γ(t) and

\frac{∂α(t, u)}{∂u}|_{u=0} = X(t)

for all t ∈ [a, b]. Moreover if X(a) = 0 and X(b) = 0 then the map α may be chosen such that α(a, u) = γ(a) and α(b, u) = γ(b) for all u ∈ (−ε, ε).

Proof A map α: I × (−ε, ε) → M with the required property is given by

α(t, u) = exp_{σ(t)}(uX(t))

(so that, for any t ∈ I, u ↦ α(t, u) is the unique geodesic in M with velocity vector X(t) at u = 0). Standard existence and smoothness theorems for solutions of systems of ordinary differential equations can be used to show that the map α is well-defined and smooth for some sufficiently small value of ε.

6.1 Length-Minimizing Curves in Riemannian Manifolds

Let (M, g) be a Riemannian manifold, and let γ: [a, b] → M be a smooth curve in M. The length L(γ) of γ is defined by

L(γ) = ∫_{a}^{b} |γ′(t)| dt,

where |γ′(t)|^{2} = g(γ′(t), γ′(t)).

A continuous curve γ: [a, b] → M is said to be piecewise smooth if there exists a partition {t_{0}, t_{1}, …, t_{k}} of the interval [a, b], where a = t_{0} < t_{1} < ⋯ < t_{k} = b, such that the restriction γ|[t_{i−1}, t_{i}] of γ to the interval [t_{i−1}, t_{i}] is smooth for i = 1, 2, …, k. The length L(γ) of γ is the sum of the lengths of the smooth curves γ|[t_{i−1}, t_{i}] for i = 1, 2, …, k.

Lemma 6.3 Let (M, g) be a Riemannian manifold, let α: [a, b] × (−ε, ε) → M be a smooth map, and let γ: [a, b] → M and α_{u}: [a, b] → M be the smooth curves in M defined by γ(t) = α(t, 0) and α_{u}(t) = α(t, u) for all t ∈ [a, b] and u ∈ (−ε, ε) (so that γ = α_{0}), and let L(α_{u}) denote the length of α_{u}. Then

\frac{dL(α_{u})}{du}|_{u=0} = \frac{1}{|γ′(b)|}g(γ′(b), X(b)) − \frac{1}{|γ′(a)|}g(γ′(a), X(a))

− ∫_{a}^{b} g (\frac{D}{dt} (\frac{1}{|γ′(t)|} \frac{dγ(t)}{dt}) , X(t)) dt,

where

X(t) = \frac{∂α(t, u)}{∂u}|_{u=0} .

In particular, if γ: [a, b] → M is parameterized by arclength, then

\frac{dL(α_{u})}{du}|_{u=0} = g(γ′(b), X(b))−g(γ′(a), X(a))−∫_{a}^{b} g (\frac{D}{dt} (\frac{dγ(t)}{dt}) , X(t)) dt.

Proof Using Lemma 4.2, we see that

\frac{∂}{∂u} ∥\frac{∂α(t, u)}{∂t}∥ = \frac{∂}{∂u}g (\frac{∂α(t, u)}{∂t}, \frac{∂α(t, u)}{∂t})^{\frac{1}{2}}

= \frac{1}{2}g (\frac{∂α(t, u)}{∂t}, \frac{∂α(t, u)}{∂t})^{− \frac{1}{2}} \frac{∂}{∂u}g (\frac{∂α(t, u)}{∂t}, \frac{∂α(t, u)}{∂t})
= ∥\frac{∂α(t, u)}{∂t}∥^{−1} g (\frac{∂α(t, u)}{∂t}, \frac{D}{du} \frac{∂α(t, u)}{∂t})
= ∥\frac{∂α(t, u)}{∂t}∥^{−1} g (\frac{∂α(t, u)}{∂t}, \frac{D}{dt} \frac{∂α(t, u)}{∂u})

Thus

\frac{dL(α_{u})}{du}|_{u=0} = ∫_{a}^{b} \frac{1}{|γ′(t)|}g (γ′(t), \frac{DX(t)}{dt}) dt
= ∫_{a}^{b} \frac{d}{dt} (\frac{1}{|γ′(t)|}g(γ′(t), X(t))) dt

− ∫_{a}^{b} g (\frac{D}{dt} (\frac{1}{|γ′(t)|} \frac{dγ(t)}{dt}) , X(t)) dt

= \frac{1}{|γ′(b)|}g(γ′(b), X(b)) − \frac{1}{|γ′(a)|}g(γ′(a), X(a))

− ∫_{a}^{b} g (\frac{D}{dt} (\frac{1}{|γ′(t)|} \frac{dγ(t)}{dt}) , X(t)) dt,

as required.

Theorem 6.4 Let p and q be distinct points in a Riemannian manifold (M, g), and let γ: [a, b] → M be a piecewise smooth curve in M from p to q, parameterized by arclength. Suppose that the length of γ is less than or equal to the length of every other piecewise smooth curve from p to q. Then γ is a smooth (unbroken) geodesic in M.

Proof First consider the case when γ: [a, b] → M is smooth. Choose a smooth function f: [a, b] → ℝ for which f(a) = 0, f(b) = 0 and f(t) > 0 for all t ∈ (a, b), and let X be the smooth vector field along γ given by

X(t) = f(t)\frac{D}{dt} \frac{dγ(t)}{dt}.

Then there exists a smooth map α: [a, b]×(−ε, ε) → M such that α(a, u) = p and α(b, u) = q for all u, and

\frac{∂α(t, u)}{∂u}|_{u=0} = X(t)

for all t ∈ [a, b] (see Lemma 6.2). Let α_{u}: [a, b] → M be given by α_{u}(t) = α(t, u). It follows from Lemma 6.3 that

0 = \frac{dL(α_{u})}{du}|_{u=0} = − ∫_{a}^{b} g (\frac{D}{dt} (\frac{dγ(t)}{dt}) , X(t)) dt
= − ∫_{a}^{b} f(t) ∥\frac{D}{dt} (\frac{dγ(t)}{dt})∥^{2} dt.

But the integrand is everywhere non-negative. We deduce that

f(t) ∥\frac{D}{dt} (\frac{dγ(t)}{dt})∥^{2} dt = 0,

for all t ∈ [a, b], and thus

\frac{D}{dt} (\frac{dγ(t)}{dt}) = 0

for all t ∈ [a, b], showing that γ: [a, b] → M is a geodesic.

Now suppose that γ is piecewise smooth. Then there exists a partition {t_{0}, t_{1}, …, t_{k}} of [a, b], where

a = t_{0} < t_{1} < ⋯ < t_{k} = b

such that γ_{i}: [t_{i−1}, t_{i}] → M is smooth for i = 1, 2, …, k, where γ_{i} = γ|[t_{i−1}, t_{i}]. Now γ_{i} must minimize length amongst all smooth curves from γ(t_{i−1}) to γ(t_{i}). It follows from the result already proved that γ_{i} is a geodesic for all i.

Choose V_{i} ∈ T_{γ(t_{i})}M for i = 1, 2, …, k − 1. One can then construct a continuous vector field X along γ with the properties that X(a) = 0, X(b) = 0, X(t_{i}) = V_{i} for i = 1, 2, …, k − 1 and X|[t_{i−1}, t_{i}] is smooth along γ_{i}. It follows, as in the proof of Lemma 6.2, that there exists a continuous map α: [a, b] × (−ε, ε) → M for some ε > 0 satisfying

\frac{∂α(t, u)}{∂u}|_{u=0} = X(t)

for all t ∈ [a, b], where α|[t_{i−1}, t_{i}] × (−ε, ε) is smooth for all i. For each u ∈ (−ε, ε) let α_{u}: [a, b] → M be the piecewise smooth curve given by α_{u}(t) = α(t, u). Now γ_{i} is a geodesic for i = 1, 2, …, k. It follows from Lemma 6.3 that

\frac{dL(α_{u})}{du}|_{u=0} = ∑_{i=1}^{k} (g(γ′_{i}(t_{i}), V_{i}) − g(γ′_{i}(t_{i−1}), V_{i−1}))

= ∑_{i=1}^{k−1} g(γ′_{i}(t_{i}) − γ′_{i+1}(t_{i}), V_{i}),

where V_{0} ∈ T_{p}M and V_{k} ∈ T_{q}M are given by V_{0} = 0 and V_{k} = 0. But γ = α_{0}, and γ minimizes length amongst all piecewise smooth curves from p to q. Therefore

∑_{i=1}^{k−1} g(γ′_{i}(t_{i}) − γ′_{i+1}(t_{i}), V_{i}) = 0

for all possible choices of V_{1}, V_{2}, …, V_{k−1}, and thus γ′_{i}(t_{i}) = γ′_{i+1}(t_{i}) for all i. We deduce that γ_{i} and γ_{i+1} are portions of the unique maximal geodesic in M which passes through γ(t_{i}) at time t = t_{i} with velocity vector γ′_{i}(t_{i}). We deduce from this that the curve γ is smooth around t_{i} for all i, and is thus a smooth (unbroken) geodesic from p to q, as required.

6.2 Geodesic Spheres and Gauss’ Lemma

Let (M, g) be a Riemannian manifold, and let m be a point of M. Now Lemma 6.1 ensures that some open neighbourhood of zero in the tangent space T_{m}M is mapped diffeomorphically onto an open set in M under the exponential map. We define the injectivity radius δ_{m} of M at the point m as follows: if the exponential map can be defined over the whole of the tangent space T_{m}M at m then we set δ_{m} = +∞; otherwise we define δ_{m} to be the supremum of all positive real numbers r with the property that the exponential map exp_{m} maps the open ball of radius r about the origin diffeomorphically onto some open set in M. It is not difficult to verify that the open ball of radius δ_{m} about zero in the tangent space T_{m}M is itself mapped diffeomorphically onto an open set in M, where δ_{m} is the injectivity radius of M at m.

Let (M, g) be a Riemannian manifold, and let m be a point of M. Given any real number r satisfying 0 < r < δ_{m}, where δ_{m} is the injectivity radius of M at m, the geodesic sphere S(m, r) of radius r about the point m is defined by

S(m, r) = exp_{m} ({V ∈ T_{m}M : ∥V ∥ = r}) ,

where ∥V ∥^{2} = g(V, V). (Thus S(m, r) is the set of all points in M which lie at a distance r from m along some geodesic radiating from the point m.) Note that the geodesic sphere S(m, r) is the image under the exponential map exp_{m} of the sphere of radius r about the origin in the Euclidean space T_{m}M. It follows from this that S(m, r) is a smooth submanifold of M diffeomorphic to the (n−1)-sphere S^{n−1} for all sufficiently small r, where n is the dimension of M.

The following result is known as Gauss’ Lemma. (It was proved by Gauss in his treatise Disquisitiones generales circa superficies curvas, published in 1827, in the case when M is a smooth surface in ℝ^{3}.)

Lemma 6.5 (Gauss’ Lemma) Let (M, g) be a Riemannian manifold, and let m be a point of M. Then, for all sufficiently small r, every geodesic passing through m intersects the geodesic sphere S(m, r) orthogonally (i.e., the velocity vector of the geodesic at the point of intersection is orthogonal to the tangent space to the geodesic sphere S(m, r)).

Proof Choose r such that r is less than the injectivity radius of M at the point m. Let V be a vector of length r in T_{m}M, for some sufficiently small positive number r, let γ: [0, 1] → M be the geodesic with initial conditions γ(0) = m and γ′(0) = V, and let q = γ(1). Then q ∈ S(m, r).

Let W be any vector tangent to the geodesic sphere S(m, r) at the point q. Now the sphere of radius r about the origin in T_{m}M is mapped diffeomorphically onto S(m, r) by the exponential map exp_{m}. Therefore Then W = (exp_{m} ∘σ)′(0) for some smooth curve σ: (−ε, ε) → T_{m}M with the property that σ(0) = V and ∥σ(u)∥ = r for all u ∈ (−ε, ε). Define α: [0, 1] × (−ε, ε) → M by α(t, u) = exp_{m}(tσ(u)) for all t ∈ [0, 1] and u ∈ (−ε, ε). Then α(t, 0) = γ(t) (where γ is the geodesic joining m and q), and, for each u ∈ (−ε, ε), the curve t ↦ α(t, u) is a geodesic in M joining m to exp_{m}(σ(u)) whose velocity vector at t = 0 is σ(u). Thus

\frac{D}{∂t} \frac{∂α}{∂t} = 0,

and

g (\frac{∂α(t, u)}{∂t}, \frac{∂α(t, u)}{∂t}) = g(σ(u), σ(u)) = r^{2}

for all t ∈ [0, 1] and u ∈ (−ε, ε), since the length of the velocity vector is constant along any geodesic. But

\frac{D}{∂u} \frac{∂α}{∂t} = \frac{D}{∂t} \frac{∂α}{∂u},

by Lemma 3.6 (since the Levi-Civita connection is torsion-free by definition). Therefore

\frac{∂}{∂t}g (\frac{∂α}{∂t}, \frac{∂α}{∂u}) = g (\frac{D}{∂t} \frac{∂α}{∂t}, \frac{∂α}{∂u}) + g (\frac{∂α}{∂t}, \frac{D}{∂t} \frac{∂α}{∂u})
= g (\frac{∂α}{∂t}, \frac{D}{∂t} \frac{∂α}{∂u}) = g (\frac{∂α}{∂t}, \frac{D}{∂u} \frac{∂α}{∂t})
= \frac{1}{2} \frac{∂}{∂u}g (\frac{∂α}{∂t}, \frac{∂α}{∂t}) = 0.

But ∂α/∂u = 0 when t = 0 (since α(0, u) = m for all u). Therefore

g (\frac{∂α}{∂t}, \frac{∂α}{∂u}) = 0

for all t ∈ [0, 1] and u ∈ (−ε, ε). In particular, if we set t = 1 and u = 0 then we deduce that W ∈ T_{q}S(m, r) is orthogonal to γ′(1), as required.

Lemma 6.6 Let (M, g) be a Riemannian manifold, let m be a point of M, and let S(m, r) be the geodesic sphere of radius r about the point m. If r is chosen sufficiently small then the length L(σ) of every piecewise smooth curve σ in M from m to a point on S(m, r) satisfies L(σ) ≥ r.

Proof Let r be chosen such that r is less than the injectivity radius of M at the point m. It suffices to prove that L(σ) ≥ r for every piecewise smooth curve σ in the closed ball about m bounded by the geodesic sphere S(m, r) (since it then follows that any piecewise smooth path leaving this ball must cross S(m, r) and thus must have length greater than r). Let σ: [a, b] → M be a piecewise smooth path contained in this closed ball. Without loss of generality we may assume that σ(t) ≠ m for all t ∈ (a, b]. We can then write σ(t) = exp_{m}(u(t)θ(t)) for all t ∈ [a, b], where u: [a, b] → [0, +∞) is a piecewise smooth non-negative function on [a, b] and θ: [a, b] → T_{m}M is a piecewise smooth curve contained in the unit sphere of T_{m}M (so that ∥θ(t)∥ = 1 for all t ∈ [a, b]). Now

\frac{d}{dt}(u(t)θ(t)) = u′(t)θ(t) + u(t)θ′(t),

But the vectors θ(t) and u(t)θ′(t) in the tangent space to T_{m}M at u(t)θ(t) are tangent to the curves s ↦ sθ(t) and s ↦ u(t)θ(s) at at s = u(t) and s = t respectively. It follows that

σ′(t) = u′(t)N(t) + V (t),

where N(t) is the unit vector tangent to the geodesic s ↦ exp_{m}(sθ(t)) at s = u(t) and V (t) is tangent to the curve s ↦ exp_{m}(u(t)θ(s)) at s = t. But exp_{m}(u(t)θ(s)) belongs to the geodesic sphere S(m, u(t)) of radius u(t) about m for all s, and hence the vector V (t) is tangent to the geodesic sphere S(m, u(t)) at σ(t). It follows from Gauss’ Lemma (Lemma 6.5) that the vectors N(t) and V (t) are orthogonal, so that

∥σ′(t)∥^{2} = g(σ′(t), σ′(t)) = u′(t)^{2}g(N(t), N(t)) + g(V (t), V (t)) ≥= |u′(t)|^{2}.

Therefore

L(σ) = ∫_{a}^{b} ∥σ′(t)∥ dt ≥ ∫_{a}^{b} |u′(t)| dt ≥ |u(b) − u(a)| = r.

as required.

7 Complete Riemannian Manifolds

Let (M, g) be a Riemannian manifold. We say that this Riemannian manifold is geodesically complete if and only if every geodesic in M can be extended to a geodesic γ: ℝ → M defined over the whole of ℝ.

Note that if (M, g) is a geodesically complete Riemannian manifold then, for each point m of M, the exponential map exp_{m}: T_{m}M → M is defined over the whole of the tangent space T_{m}M to M at m.

Given points p and q in a connected Riemannian manifold (M, g) we define the Riemannian distance d(p, q) from p to q to be the infimum (i.e., the greatest lower bound) of the lengths of all piecewise smooth curves starting at p and ending at q. Note that d(p, q) ≥ 0 and d(p, q) = d(q, p). The Triangle Inequality

d(m, q) ≤ d(m, p) + d(p, q)

is clearly satisfied for all m, p, q ∈ M (since every piecewise smooth curve from m to p combines with every piecewise smooth curve from p to q to give a piecewise smooth curve from m to q). If p ≠ q then d(p, q) > 0. Indeed there exists some positive real number r such that the closed ball about p bounded by the geodesic sphere S(p, r) of radius r about p does not contain the point q. It follows from Lemma 6.6 that the length of every piecewise smooth curve from p to q must exceed r, since the curve must cross S(p, r). We deduce that every connected Riemannian manifold (M, g) is a metric space with respect to the Riemannian distance function. Moreover it follows easily from Lemma 6.1 and Lemma 6.6 that the topology generated by this distance function is the given topology on M.

The following results are due to Hopf and Rinow.

Theorem 7.1 Let (M, g) be a connected Riemannian manifold and let m be a point of M. Suppose that the exponential map exp_{m}: T_{m}M → M at m is defined over the whole of the tangent space T_{m}M to M at m (i.e., every geodesic passing through the point m can be extended to a geodesic t ↦ γ(t) defined for all t). Then, given any q ∈ M, there exists a geodesic from m to q which minimizes length amongst all piecewise smooth curves from m to q.

Proof Let q be a point of M which is distinct from m, and let r denote the Riemannian distance d(m, q) between the points m and q. Choose δ > 0 such that δ is less than the injectivity radius of M at m and q does not belong to the closed ball about m in M bounded by the geodesic sphere S(m, δ) of radius δ about m. Then every piecewise smooth curve from m to q must cross the geodesic sphere S(m, δ). It follows from the compactness of S(m, δ) that there exists some tangent vector V ∈ T_{m}M of unit length at m with the property that

d(exp_{m}(δV), q) = inf{d(s, q) : s ∈ S(m, δ)}.

Let γ: ℝ → M be the geodesic, parameterized by arclength, defined by γ(t) = exp_{m}(tV). We claim that γ(r) = q, where r = d(m, q).

Now every path from m to q must cross the geodesic sphere S(m, δ) of radius δ about m. But d(m, q) is defined to be the infimum of the lengths of all smooth curves from m to q. Therefore

d(m, q) = δ + inf{d(p, q) : p ∈ S(m, δ)} = δ + d(γ(δ), q)

(since every piecewise smooth curve from m to q must cross S(m, δ), and d(m, p) = δ for all p ∈ S(m, δ)). Thus d(γ(δ), q) = r − δ, where r = d(m, q). Let

u = sup{t ∈ [δ, r] : d(γ(t), q) = r − t},

where r = d(m, q). We shall show that u = r.

Now follows from the continuity of the Riemannian distance function that d(γ(u), q) = r − u. Suppose that it were the case that u < r. We show that this would lead to a contradiction. For if u < r then we could find some sufficiently small η > 0 with the property that every path from γ(u) to q must cross the geodesic sphere S(γ(u), η) of radius η about γ(u). But then there would exist p ∈ S(γ(u), η) such that

d(p, q) = inf{d(p′, q) : p′ ∈ S(γ(u), η)},

since S(γ(u), η) is compact. Moreover

d(γ(u), q) = η + inf{d(p′, q) : p′ ∈ S(γ(u), η)} = η + d(p, q),

so that d(p, q) = r − u − η. Thus

r = d(m, q) ≤ d(m, p) + d(p, q) ≤ d(m, p) + r − u − η,

and hence d(m, p) ≥ u + η. But if α: [0, u + η] → M is the piecewise smooth curve from m to p, parameterized by arclength, obtained by concatenating the geodesic γ from m to γ(u) with the geodesic of length η from γ(u) to p, then the length of the curve α is u + η, and therefore α minimized length amongst all piecewise smooth curves from m to p. It follows from Lemma 6.4 that α is a smooth (unbroken) geodesic, so that α(t) = γ(t) for all t ∈ [0, u + η]. But then p = γ(u + η) so that d(γ(u + η), q) = r − u − η, contradicting the definition of u. Thus excludes the possibility that u < r. Therefore u = r, and hence d(γ(r), q) = 0. Thus q = γ(r), as required.

Corollary 7.2 Let (M, g) be a connected Riemannian manifold. Then the following three conditions are equivalent:

(i) the Riemannian distance function on M is complete (i.e., every Cauchy sequence in M converges),

(ii) the Riemannian manifold (M, g) is geodesically complete,

(iii) there exists a point m of M with the property that the exponential map exp_{m} is defined over the whole of the tangent space T_{m}M to M at m (i.e., every geodesic passing through the point m can be extended to a geodesic from ℝ into M).

Proof First we show that (i) implies (ii). Thus suppose that the Riemannian distance function of M is complete. Let γ: I → M be a maximal geodesic in M, defined on an open interval I in ℝ. We must show that I = ℝ. Suppose that it were the case the the interval I is bounded above. Let u = sup I. Then we could find an ascending sequence (t_{j} : j ∈ ℕ) of elements of I which converges to u. Now the length ∥γ′(t)∥ of the velocity vector γ′(t) of γ is constant along the geodesic. Let ∥γ′(t)∥ = C. Then d(γ(t_{j}), γ(t_{k})) ≤ C(t_{k} − t_{j}) for all natural numbers j and k (where d(γ(t_{j}), γ(t_{k})) is the Riemannian distance from t_{j} to t_{k}). But the sequence (t_{j} : j ∈ ℕ) is a Cauchy sequence in ℝ. It follows that the sequence (γ(t_{j}) : j ∈ ℕ) is a Cauchy sequence in M and therefore converges to some point m of M, since M is complete with respect to the Riemannian distance function.

Now it follows from standard existence theorems for solutions of differential equations, applied to the second order system characterizing geodesics in M with respect to some smooth coordinate system, that there exists some open neighbourhood U of m and some δ > 0 such that exp_{q}(V) is welldefined for all q ∈ U and V ∈ T_{q}M satisfying ∥V ∥ < δ. Let j be chosen large enough to ensure that t_{j} ∈ U and u − t_{j} < ε, where ε = δ/(2C). Then ∥(t−t_{j})γ′(t_{j})∥ < δ for all t ∈ (t_{j} −ε, t_{j} +ε). Thus if σ(t) = exp_{q}((t−t_{j})γ′(t_{j})) for all t ∈ (t_{j} − ε, t_{j} + ε) then σ: (t_{j} − ε, t_{j} + ε) is a geodesic in M, and σ′(t_{j}) = γ′(t_{j}). We deduce that σ(t) = γ(t) for all t sufficiently close to t_{j}. It follows from the maximality of γ that (t_{j} − ε, t_{j} + ε) ⊂ I (since otherwise we could extend γ be defining γ(t) = σ(t) for all t ∈ (t_{j} − ε, t_{j} + ε) \ I). But this is impossible since t_{j} + ε > u, where u = sup I. We deduce that the interval I is not bounded above. Similarly the interval I is not bounded below. Therefore I = ℝ. This shows that (i) implies (ii).

A Riemannian manifold (M, g) is geodesically complete if and only if, for every m ∈ M, the exponential map exp_{m}: T_{m}M → M is defined over the whole of the tangent space T_{m}M at m. Thus (ii) implies (iii).

Finally we show that (iii) implies (i). Suppose that (iii) holds. Let m be a point of M with the property that the exponential map exp_{m} is defined over the whole of the tangent space T_{m}M to M at m. Let (m_{j} : j ∈ ℕ) be a Cauchy sequence in M. Then d(m, m_{j}) remains bounded as j → +∞. Therefore there exists some R > 0 with the property that d(m, m_{j}) ≤ R. Now it follows from Theorem 7.1 that each point m_{j} can be joined to m by a geodesic of length d(m, m_{j}). Therefore m_{j} ∈ B(m, R) for all j, where

B(m, R) = exp_{m} ({V ∈ T_{m}M : |V | ≤ R}) .

Now B(m, R) is compact, since it is the image of a compact subset of T_{m}M under the continuous map exp_{m}: T_{m}M → M. Moreover any compact metric space is complete. Therefore the Cauchy sequence m_{1}, m_{2}, m_{3} converges (in M) to some point of B(m, R). This shows that M is complete. Thus (iii) implies (i). Thus conditions (i), (ii) and (iii) are equivalent, as required.

We see that a connected Riemannian manifold is geodesically complete if and only if it is a complete metric space with respect to the Riemannian distance function. Such a Riemannian manifold is said to be complete.

7.1 Local Isometries and Covering Maps

Definition Let (M_{1}, g_{1}) and (M_{2}, g_{2}) be Riemannian manifolds of the same dimension, and let φ: M_{1} → M_{2} be a smooth map. We say that φ is a local isometry if and only if

g_{2}(φ_{∗}X, φ_{∗}Y) = g_{1}(X, Y)

for all m ∈ M_{1} and X, Y ∈ T_{m}M_{1}, where φ_{∗}: T_{m}M_{1} → T_{φ(m)}M_{2} is the derivative of the map φ at m. A local isometry φ: M_{1} → M_{2} is said to be an isometry if it is also a diffeomorphism from M_{1} to M_{2}.

Observe that if φ: M_{1} → M_{2} is a local isometry then ∥φ_{∗}X∥ = ∥X∥ for all tangent vectors X. It follows that the derivative φ_{∗}: T_{m}M_{1} → T_{φ(m)}M_{2} of φ at each point m of M_{1} is an injective homomorphism between vector spaces of the same dimension and is thus an isomorphism of vector spaces. It follows from the Inverse Function Theorem that φ maps some open neighbourhood of each point m of M_{1} diffeomorphically onto an open neighbourhood of φ(m) in M_{2}. Moreover the Levi-Civita connection and the Riemann curvature tensor around a point in a Riemannian manifold is completely determined by the Riemannian metric around that point. It follows that the Levi-Civita connections and Riemann curvature tensors of the Riemannian manifolds (M_{1}, g_{1}) and (M_{2}, g_{2}) correspond under any local isomorphism φ: M_{1} → M_{2}. Moreover a smooth curve γ: I → M_{1} is a geodesic in M_{1} if and only if φ: γ: I → M_{2} is a geodesic in M_{2}.

We now recall the definition of covering maps. Let X̃ and X be topological spaces and let φ: X̃ → X be a continuous map. An open set U in X is said to be evenly covered by the map φ: X̃ → X if the preimage φ^{−1}(U) of U is a disjoint union of open sets in X̃, each of which is mapped homeomorphically onto U by the map φ. The map φ: X̃ → X is said to be a covering map if it is surjective and each point of X has an open neighbourhood evenly covered by the map φ. If the topological space X is simply-connected then any covering map φ: X̃ → X is a homeomorphism.

Theorem 7.3 Let (M, g) and (M̃, g̃) be connected (non-empty) Riemannian manifolds of the same dimension and let φ: M̃ → M be a local isometry. Suppose that M̃ is complete. Then M is complete and the map φ: M̃ → M is a covering map. In particular, φ: M̃ → M is surjective.

Proof We first prove the following lifting property for geodesics in M: given any geodesic γ: I → M, defined on some interval I in ℝ, and given t_{0} ∈ I and q ∈ M̃ satisfying φ(q) = γ(t_{0}), there exists a unique geodesic γ̃: I → M̃ such that γ̃(t_{0}) = q and φ ∘ γ̃ = γ. Indeed the derivative φ_{∗} of φ at q is an isomorphism of vector spaces, and hence there exists a unique tangent vector V ∈ T_{q}M̃ at q satisfying φ_{∗}V = γ′(t_{0}). Moreover there exists a geodesic η: ℝ → M̃ in M̃ satisfying η(t_{0}) = q and η′(t_{0}) = V, since M̃ is complete. Then φ ∘ η: ℝ → M is a geodesic in M, and (φ ∘ η)′(t_{0}) = γ′(t_{0}). But the geodesic γ is uniquely determined by γ′(t_{0}). We deduce that γ = (φ ∘ η)|I. Thus if we define γ̃: I → M̃ to be the restriction η|I of η: ℝ → M to the interval I then γ̃ is the required lift of the geodesic γ to M̃. We deduce also that any geodesic passing through a point of the image φ(M̃) of the map φ: M̃ → M can be extended to a geodesic defined over the whole of ℝ. We deduce from Theorem 7.1 that the exponential map exp_{m}: T_{m}M → M of M at a point m of φ(M) is surjective, and, since every geodesic passing through m is of the form φ ∘ γ̃ for some geodesic γ̃ in M̃, it follows that φ(M̃) = M. We also deduce that every geodesic in M can be extended to a geodesic defined over the whole of ℝ, so that M is complete.

Let m be a point in M. It follows from Lemma 6.1 that there exists some δ_{m} > 0 such that the exponential map exp_{m} at m maps the open ball of radius δ about the zero vector in the tangent space T_{m}M to M at m diffeomorphically onto an open neighbourhood U of m. Then every point of U can be joined to m by a unique geodesic in U of length less than δ_{m}. We show that U is evenly covered by the map φ.

Let p be an element of M̃ satisfying φ(p) = m and let U_{p} be the subset of φ^{−1}(U) consisting of all points that can be joined to p by a geodesic contained wholly in φ^{−1}(U). Now if γ̃ is any geodesic in φ^{−1}(U) joining p to a point q in U_{p} then φ ∘ γ̃ is a geodesic in U joining m to φ(q). Conversely, given any geodesic γ in U joining m to a point u of U, there exists a unique geodesic γ̃ in U_{p} joining p to some point q of U_{p} satisfying φ(q) = u. Moreover there is exactly one point q of U_{p} satisfying φ(q) = u, since the geodesic γ is the unique geodesic in U joining m to u, and therefore γ̃ is the unique geodesic in U_{p} joining p to a point of φ^{−1}({u}). We deduce that U_{p} is mapped bijectively on U under the map φ. But φ: M̃ → M is a local diffeomorphism. It follows easily that U_{p} is mapped diffeomorphically onto U under the map φ.

Now let q be any element of φ^{−1}(U). Then φ(q) can be joined to m by a unique geodesic γ contained wholly in U. The geodesic γ lifts to a geodesic γ̃ in φ^{−1}(U) joining q to some element p of M̃ satisfying φ(p) = m, where φ ∘ γ̃ = γ. Moreover p is uniquely determined by q, since any geodesic in φ^{−1}(U) joining q to an element of φ^{−1}({m}) is mapped under φ to the unique geodesic in U joining φ(q) to m. We deduce that φ^{−1}(U) is the disjoint union of the sets U_{p}, as p ranges over the elements of φ^{−1}({m}), and thus U is evenly covered by the map φ. This shows that φ: M̃ → M is a covering map, as required.

We recall that if X is a simply-connected topological space then any covering map φ: X̃ → X is a homeomorphism. Also any local isometry is a local diffeomorphism, and therefore any local isometry which also a homeomorphism is a diffeomorphism, and is thus an isometry of Riemannian manifolds. The follows result therefore follows directly from Theorem 7.3.

Corollary 7.4 Let (M, g) and (M̃, g̃) be connected (non-empty) Riemannian manifolds of the same dimension and let φ: M̃ → M be a local isometry. Suppose that M̃ is complete and that M is simply-connected. Then the map φ: M̃ → M is an isometry of Riemannian manifolds.

8 Jacobi Fields

Let (M, g) be a Riemannian manifold, and let γ: [a, b] → M be a geodesic in M. A Jacobi field along γ is a vector field V along γ which satisfies the Jacobi equation

\frac{D^{2}V (t)}{dt^{2}} = R(γ′(t), V (t))γ′(t),

where R denotes the curvature tensor of the Levi-Civita connection on M. First we show that Jacobi fields arise naturally from variations of the geodesic γ through neighbouring geodesics.

Lemma 8.1 Let γ: I → M be a geodesic in a Riemannian manifold (M, g) and let

α: I × (−ε, ε) → M

be a smooth map satisfying α(t, 0) = γ(t) for all t ∈ I. Let V be the vector field along the geodesic γ defined by

V (t) = \frac{∂α(t, u)}{∂u}|_{u=0} .

Suppose that, for each u ∈ (−ε, ε), the curve t ↦ α(t, u) is a geodesic in M. Then the vector field V satisfies the Jacobi equation

\frac{D^{2}V (t)}{dt^{2}} = R(γ′(t), V (t))γ′(t).

Proof First we note that

\frac{D}{∂t} \frac{∂α}{∂t} = 0,

since each curve t ↦ α(t, u) is a geodesic Also

\frac{D}{∂t} \frac{D}{∂u} \frac{∂α}{∂t} − \frac{D}{∂u} \frac{D}{∂t} \frac{∂α}{∂t} = R(\frac{∂α}{∂t}, \frac{∂α}{∂u})\frac{∂α}{∂t}

and

\frac{D}{∂t} \frac{∂α}{∂u} = \frac{D}{∂u} \frac{∂α}{∂t}

by Lemma 3.6, using the fact that the Levi-Civita connection is torsion-free. Therefore

\frac{D^{2}}{∂t^{2}} \frac{∂α}{∂u} = \frac{D}{∂t} \frac{D}{∂u} \frac{∂α}{∂t}
= R(\frac{∂α}{∂t}, \frac{∂α}{∂u})\frac{∂α}{∂t} + \frac{D}{∂u} \frac{D}{∂t} \frac{∂α}{∂t}
= R(\frac{∂α}{∂t}, \frac{∂α}{∂u})\frac{∂α}{∂t}.

Now

\frac{∂α(t, u)}{∂t}|_{u=0} = γ′(t), \frac{∂α(t, u)}{∂u}|_{u=0} = V (t).

Thus, on setting u = 0, we deduce that

\frac{D^{2}V (t)}{dt^{2}} = R(γ′(t), V (t))γ′(t),

as required.

The next result shows how Jacobi fields can be used to study the derivative of the exponential map.

Lemma 8.2 Let (M, g) be a Riemannian manifold, let γ: [0, 1] → M be a geodesic in M defined on the interval [0, 1], let m = γ(0), and let X = γ′(0). Then exp_{m∗}(Y_{X}) = V (1) for any Y ∈ T_{m}M, where

exp_{m∗}: T_{X}(T_{m}M) → T_{γ(1)}M

is the derivative of the exponential map at X, Y_{X} ∈ T_{X}(T_{m}M) is the velocity vector of the curve u ↦ X + uY at u = 0, and V is the Jacobi field along γ satisfying the initial conditions

V (0) = 0, \frac{DV (t)}{dt}|_{t=0} = Y.

Proof Let α: [0, 1]×(−ε, ε) → M be the map defined by α(t, u) = exp_{m}(tX+ tuY) for all t ∈ [0, 1] and u ∈ (−ε, ε), where ε > 0 is chosen small enough to ensure that the map α is well-defined. For fixed u, the curve t ↦ α(t, u) is a geodesic in M. It follows from Lemma 8.1 that the vector field V along γ defined by

V (t) = \frac{∂α(t, u)}{∂u}|_{u=0}

is a Jacobi field along γ. Moreover

\frac{DV (t)}{dt}|_{t=0} = \frac{D}{∂t} \frac{∂α}{∂u}|_{(t,u)=(0,0)} = \frac{D}{∂u} \frac{∂α}{∂t}|_{(t,u)=(0,0)} = \frac{D}{∂u}(X + uY)|_{u=0} = Y,

(where we have used Lemma 3.6), and

V (1) = \frac{dα(1, u)}{du}|_{u=0} = \frac{d}{du} exp_{m}(X + uY)|_{u=0}
= exp_{m∗}( \frac{d}{du}(X + uY)|_{u=0}) = exp_{m∗}(Y_{X}),

as required.

8.1 Flat Riemannian manifolds

Now let us consider the case when the Riemann curvature tensor of a Riemannian manifold (M, g) vanishes everywhere on M. Such a Riemannian manifold is said to be flat.

The tangent space T_{m}M to a Riemannian manifold (M, g) has a natural flat Riemannian metric corresponding to the inner product on T_{m}M given by the Riemannian metric.

Theorem 8.3 Let (M, g) be a flat Riemannian manifold, and let m be a point of M. Then the exponential map exp_{m}: U → M is a local isometry, where U ⊂ T_{m}M is the domain of the exponential map exp_{m} at the point m.

Proof Let X be an element of U, and let Y and Z be tangent vectors at m. Let γ: ℝ → M be the geodesic defined by γ(t) = exp_{m}(tX) for all t ∈ ℝ, and let fields t ↦ Y (t) and t ↦ Z(t) be the parallel vector fields along γ satisfying Y (0) = Y, Z(0) = Z,

\frac{DY (t)}{dt} = 0 and \frac{DZ(t)}{dt} = 0.

Then

\frac{D^{2}}{dt^{2}} (tY (t)) = 0, \frac{D^{2}}{dt^{2}} (tZ(t)) = 0.

But the curvature tensor of M is zero everywhere on M, since M is flat, and therefore the vector fields t ↦ tY (t) and t ↦ tZ(t) satisfy the Jacobi equation along γ. Thus exp_{m∗} Y_{X} = Y (1) and exp_{m∗} Z_{X} = Z(1) by Lemma 8.2, where Y_{X} ∈ T_{X}(T_{m}M) and Z_{X} ∈ (T_{m}M) are tangent to the curves u ↦ X + uY and u ↦ X + uZ at u = 0. But

\frac{d}{dt}g(Y (t), Z(t)) = g(\frac{D}{dt}Y (t), Z(t)) + g(Y (t), \frac{D}{dt}Z(t)) = 0,

and hence

g(exp_{∗} Y_{X}, exp_{∗} Z_{X}) = g(Y (1), Z(1)) = g(Y, Z) = ⟨Y_{X}, Z_{X}⟩,

where ⟨., .⟩ denotes the inner product on the tangent spaces of T_{m}M defined by the Riemannian metric g at the point m. Thus the derivative of the exponential map at X ∈ T_{m}M is an isometry from T_{X}(T_{m}M) to T_{γ(1)}M. Thus exp_{m}: U → M is a local isometry.

Corollary 8.4 Let (M, g) be a complete simply-connected Riemannian manifold of dimension n. Suppose that M is flat. Then M is isometric to the Euclidean space ℝ^{n}.

Proof If M is complete and simply-connected then the exponential map exp_{m} is defined over the whole of T_{m}M and is a local isometry. Moreover T_{m}M is complete (since any Euclidean space is a complete Riemannian manifold). It follows from Corollary 7.4 that exp_{m}: T_{m}M → M is an isometry, so that M is isometric to n-dimensional Euclidean space.

8.2 The Cartan-Hadamard Theorem

Let (M, g) be a Riemannian manifold, let m be a point of M, and let P be a 2-dimensional subspace of the tangent space T_{m}M to M at m. We recall that the sectional curvature K(P) in the plane P is given terms of the curvature tensor of M by the formula

K(P) = R(E_{1}, E_{2}, E_{1}, E_{2}) = g(E_{1}, R(E_{1}, E_{2})E_{2}),

where (E_{1}, E_{2}) is any orthonormal basis for P. We conclude that the sectional curvatures of the Riemannian manifold (M, g) are all non-positive at a point m if and only if g(X, R(X, Y)Y) ≤ 0 for all tangent vectors X and Y at m.

Theorem 8.5 (Cartan-Hadamard) Let (M, g) be a complete simply-con-nected Riemannian manifold. Suppose that the sectional curvatures K(P) of M all satisfy K(P) ≤ 0. Then M is diffeomorphic to ℝ^{n}.

Proof Choose a point m of M. Then the exponential map exp_{m}: T_{m}M → M at m is defined over the whole of the tangent space of M at m, since (M, g) is complete. Given X ∈ T_{m}M, let γ: ℝ → M be the geodesic in M given by γ(t) = exp_{m}(tX) for all t ∈ ℝ, and let t ↦ V (t) be a Jacobi field along γ. satisfying

V (0) = 0, DV (t)/dt|_{t=0} ≠ 0.

We show that t ↦ ∥V (t)∥ is an increasing function of t for t = 0, so that V (t) ≠ 0 for all t > 0. Now t ↦ V (t) satisfies the Jacobi equation

\frac{D^{2}V (t)}{dt^{2}} = R(γ′(t), V (t))γ′(t) = −R(V (t), γ′(t))γ′(t),

and hence

\frac{d^{2}}{dt^{2}} ∥V (t)∥^{2} = 2\frac{d}{dt}g (V (t), \frac{DV (t)}{dt})
= 2g (\frac{DV (t)}{dt}, \frac{DV (t)}{dt}) + 2g (V (t), \frac{D^{2}V (t)}{dt^{2}})
= 2 ∥\frac{DV (t)}{dt}∥^{2} − 2g(V (t), R(V (t), γ′(t))γ′(t)).

But g(V (t), R(V (t), γ′(t))γ′(t) ≤ 0 for all t, since the sectional curvatures of M are all non-positive. Thus

\frac{d^{2}}{dt^{2}} ∥V (t)∥^{2} ≥ 2 ∥\frac{DV (t)}{dt}∥^{2} ≥ 0

for all t. But there exist values of t arbitrarily close to zero for which d/dt∥V (t)∥^{2} > 0, since V (0) = 0 and V (t) ≠ 0 for all sufficiently small non-zero values of t. We deduce that t ↦ ∥V (t)∥ is an increasing function for t > 0, and thus ∥V (t)∥ > 0 for all t > 0. Now it follows from Lemma 8.2 that the derivative exp_{m∗} of exp_{m}: T_{m}M → M at X sends Y_{X} to V (1), where

Y = \frac{DV (t)}{dt}|_{t=0} ,

and Y_{X} ∈ T_{X}(T_{m}M) is the velocity to u ↦ X + uY at u = 0. We deduce that exp_{m∗}: T_{X}(T_{m}M) → T_{exp_{m}(X)}M is injective and is thus an isomorphism of vector spaces.

Let g̃ be the Riemannian metric on T_{m}M defined at X ∈ T_{m}M by g̃(Y, Z) = g(exp_{m∗} Y, g(exp_{m∗} Z) for all Y, Z ∈ T_{X}(T_{m}M). The metric tensor g̃ is positive-definite at X since the derivative exp_{m∗} of exp_{m} at X is an isomorphism of vector spaces. Then exp_{m}: T_{m}M → M is a local isometry between the Riemannian manifolds (T_{m}, g̃) and (M, g).

We claim that the Riemannian manifold (T_{m}M, g̃) is complete. Now a curve γ̃ is a geodesic in T_{m}M if and only if exp_{m} ∘γ is a geodesic in M. But the exponential map exp_{m} sends straight lines through the origin in T_{m}M to geodesics in M. We conclude that every geodesic through the origin in the Riemannian manifold (T_{m}M, g̃) can be extended to a geodesic defined over the whole of the real line ℝ. It follows from Corollary 7.2 that (T_{m}M, g̃) is complete. But M is simply-connected, by hypothesis, and any local isometry from a complete Riemannian manifold to a simply-connected Riemannian manifold is an isometry (see Corollary 7.4), and is thus a diffeomorphism. We conclude that M is diffeomorphic to the Euclidean space T_{m}M, as required.

8.3 The Second Variation of Energy

Let (M, g) be a Riemannian manifold and let γ: [a, b] → M be a geodesic in M. Let α: [a, b] × (−ε, ε) → M be a smooth map with the properties that

α(t, 0) = γ(t) for all t ∈ [a, b],
α(a, u) = γ(a) for all u ∈ (−ε, ε),
α(b, u) = γ(b) for all u ∈ (−ε, ε).

Thus if α_{u}: [a, b] → M is the smooth curve defined by α_{u}(t) = α(t, u) then each α_{u} starts at γ(a) and ends at γ(b). We calculate

\frac{d^{2}E(γ(α_{u})}{du^{2}} |_{u=0} ,

where E(α_{u}) is the energy of α_{u}. Let X and Y be the smooth vector fields along the map α defined by

X(t, u) = \frac{∂α(t, u)}{∂t}, Y (t, u) = \frac{∂α(t, u)}{∂u}.

Note that Y (a, u) = 0 and Y (b, u) = 0 for all u ∈ (−ε, ε), on account of the fact that α(a, u) = γ(a) and α(b, u) = γ(b). The energy of the curve α_{u} is given by

E(α_{u}) = \frac{1}{2} ∫_{a}^{b} g(X(t, u), X(t, u)) dt.

Now

\frac{DX}{∂u} = \frac{D}{∂u} \frac{∂α}{∂t} = \frac{D}{∂t} \frac{∂α}{∂u} = \frac{DY}{∂t}

by Lemma 3.6. Thus

\frac{dE(α_{u})}{du} = ∫_{a}^{b} g (X, \frac{DX}{∂u} ) dt = ∫_{a}^{b} g (X, \frac{DY}{∂t} ) dt,

hence

\frac{d^{2}E(α_{u})}{du^{2}} = ∫_{a}^{b} (g (\frac{DX}{∂u} , \frac{DY}{∂t} ) + g (X, \frac{D}{∂u} \frac{DY}{∂t} )) dt
= ∫_{a}^{b} (g (\frac{DY}{∂t} , \frac{DY}{∂t} ) + g (X, \frac{D}{∂u} \frac{DY}{∂t} )) dt.

But

\frac{D}{∂u} \frac{DY}{∂t} = \frac{D}{∂t} \frac{DY}{∂u} + R(Y, X)Y

by Lemma 3.6. Therefore

\frac{d^{2}E(α_{u})}{du^{2}} = ∫_{a}^{b} g (\frac{DY}{∂t} , \frac{DY}{∂t} ) dt

+ ∫_{a}^{b} g (X, \frac{D}{∂t} \frac{DY}{∂u} + R(Y, X)Y ) dt.

But

∫_{a}^{b} g (X, \frac{D}{∂t} \frac{DY}{∂u} ) dt = ∫_{a}^{b} \frac{∂}{∂t} (g (X, \frac{DY}{∂u} )) dt − ∫_{a}^{b} g (\frac{DX}{∂t} , \frac{DY}{∂u} ) dt
= g (X(b, u), \frac{DY (b, u)}{∂u}) − g (X(a, u), \frac{DY (a, u)}{∂u})

− ∫_{a}^{b} g (\frac{DX}{∂t} , \frac{DY}{∂u} ) dt

= − ∫_{a}^{b} g (\frac{DX}{∂t} , \frac{DY}{∂u} ) dt,

because Y (a, u) = 0 and Y (b, u) = 0 for all u ∈ (−ε, ε). Thus

\frac{d^{2}E(α_{u})}{du^{2}} = ∫_{a}^{b} (g (\frac{DY}{∂t} , \frac{DY}{∂t} ) + g(X, R(Y, X)Y)) dt

− ∫_{a}^{b} g (\frac{DX}{∂t} , \frac{DY}{∂u} ) dt.

Now let us set u = 0. We define the vector field V along γ by

V (t) = Y (t, 0) = \frac{∂α(t, u)}{∂u}|_{u=0} .

Note that X(t, 0) = γ′(t) and

\frac{DX(t, 0)}{dt} = \frac{Dγ′(t)}{dt} = 0

(since γ is a geodesic. Therefore

\frac{d^{2}E(α_{u})}{du^{2}} |_{u=0}
= ∫_{a}^{b} (g (\frac{DV (t)}{∂t}, \frac{DV (t)}{∂t}) + g(γ′(t), R(V (t), γ′(t))V (t))) dt
= ∫_{a}^{b} (g (\frac{DV (t)}{∂t}, \frac{DV (t)}{∂t}) + R(γ′(t), V (t), V (t), γ′(t))) dt.

We can integrate the first term in this formula by parts. Using the fact that V (a) = 0 and V (b) = 0 we see that

∫_{a}^{b} g (\frac{DV (t)}{∂t}, \frac{DV (t)}{∂t}) dt = − ∫_{a}^{b} g (V (t), \frac{D^{2}V (t)}{∂t^{2}}) dt

Also

R(γ′(t), V (t), V (t), γ′(t)) = −R(V (t), γ′(t), V (t), γ′(t))
= R(V (t), γ′(t), γ′(t), V (t))
= g(V (t), R(γ′(t), V (t))γ′(t)),

by Lemma 4.4. We conclude that

\frac{d^{2}E(α_{u})}{du^{2}} |_{u=0} = ∫_{a}^{b} g (V (t), R(γ′(t), V (t))γ′(t) − \frac{D^{2}V (t)}{dt^{2}}) dt.

Thus if V is a Jacobi field along γ then

\frac{d^{2}E(α_{u})}{du^{2}} |_{u=0} = 0.

Lemma 8.6 Let (M, g) be a Riemannian manifold and let p and q be points of M. Let γ: [a, b] → M be a geodesic with γ(a) = p and γ(b) = q which minimizes length among all curves from p to q that approximate sufficiently closely to γ. Then

∫_{a}^{b} (g (V (t), \frac{D^{2}V (t)}{∂t^{2}}) + R(V (t), γ′(t), V (t), γ′(t))) dt ≤ 0

for all smooth vector fields V along γ which satisfy V (a) = 0 and V (b) = 0.

Proof Let V be a smooth vector field along γ: [a, b] → ℝ which satisfies V (a) = 0 and V (b) = 0. It follows from Lemma 6.2 that there exists a smooth map α: [a, b] × (−ε, ε) → M such that α(t, 0) = γ(t) for all t and

V (t) = \frac{∂α(t, u)}{∂u}|_{u=0} .

Moreover the map α can be chosen such that α(a, u) = p and α(b, u) = q for all u ∈ (−ε, ε). Now γ minimizes length among all smooth curves from p to q that approximate sufficiently closely to γ. Therefore the length L(α_{u}) of the curve α_{u}: [a, b] → M satisfies L(α_{u}) ≥ L(γ) for all u sufficiently close to zero (where α_{u}(t) = α(t, u) for all t ∈ [a, b]). We show that this implies that E(α(u) ≥ E(γ) for all u sufficiently close to zero. To do this we use a form of Schwarz’ Inequality which states that if f and g are functions on the closed interval [a, b] whose squares are integrable on [a, b] then

(∫_{a}^{b} f(t)g(t) dt)^{2} ≤ ∫_{a}^{b} f(t)^{2} dt ∫_{a}^{b} g(t)^{2} dt,

where equality holds if and only if the functions f and g are proportional to one another. If we apply this result where

f(t) = |α′_{u}(t)| = \sqrt{g(α′_{u}(t), α′_{u}(t))}

and where g is the constant function equal to 1 everywhere on [a, b] we see that

L(α_{u})^{2} = (∫_{a}^{b} |α′_{u}(t)| dt)^{2} ≤ (b − a) ∫_{a}^{b} |α′_{u}(t)|^{2} dt = 2(b − a)E(α_{u})

with equality if and only if |α′(t)| is constant along the curve. Now |γ′(t)| is constant along the geodesic γ, hence L(γ)^{2} = 2(b − a)E(γ). Therefore

E(α_{u}) ≥ \frac{L(α_{u})^{2}}{2(b − a)} ≥ \frac{L(γ)^{2}}{2(b − a)} = E(γ)

for all u sufficiently close to zero. We conclude that

\frac{d^{2}E(α_{u})}{du^{2}} |_{u=0} ≥ 0.

But it follows from the remarks above that

\frac{d^{2}E(α_{u})}{du^{2}} |_{u=0} = − ∫_{a}^{b} (g (V (t), \frac{D^{2}V (t)}{∂t^{2}}) + R(V (t), γ′(t), V (t), γ′(t))) dt.

The required result follows directly from this.

We can use this inequality to derive the following result.

Lemma 8.7 Let (M, g) be a Riemannian manifold all of whose sectional curvatures K(P) satisfy K(P) > π^{2}/L^{2} for some L > 0 and let γ: [0, L] → M be a geodesic in M of length L, parameterized by arclength. Then γ does not minimize length amongst all smooth curves from γ(0) to γ(L).

Proof Let E be a parallel vector field along the geodesic γ which is of unit length and which is everywhere perpendicular to γ and let V be the vector field along γ defined by

V (t) = sin (\frac{πt}{L}) E(t)

Then

\frac{D^{2}V (t)}{dt^{2}} = −\frac{π^{2}}{L^{2}} sin (\frac{πt}{L}) E(t),

so that

g (V (t), \frac{D^{2}V (t)}{dt^{2}}) = −\frac{π^{2}}{L^{2}} sin^{2} (\frac{πt}{L}) .

Also

R(V (t), γ′(t), V (t), γ′(t)) = sin^{2} (\frac{πt}{L}) R(E(t), γ′(t), E(t), γ′(t))
> \frac{π^{2}}{L^{2}} sin^{2} (\frac{πt}{L})

for all t ∈ (0, L) (since R(E(t), γ′(t), E(t), γ′(t)) is the sectional curvature of the plane spanned by the orthonormal vectors γ′(t) and E(t) and this sectional curvature exceeds π^{2}/L^{2}). Therefore

∫_{a}^{b} (g (V (t), \frac{D^{2}V (t)}{∂t^{2}}) + R(V (t), γ′(t), V (t), γ′(t))) dt > 0.

But V (0) = 0 and V (L) = 0. It follows from Lemma 8.6 that the geodesic γ: [0, L] → M is not a curve of shortest length from γ(0) to γ(L).

One can strengthen the above result. To do this we introduce the Ricci tensor ρ of the Riemannian manifold (M, g). The Ricci tensor on M is the tensor of type (0, 2) defined at the point p of M by

ρ(U, V) = ∑_{i=1}^{n} R(E_{i}, U, E_{i}, V)

for all U, V ∈ T_{p}M, where (E_{1}, E_{2}, …, E_{n}) is an orthonormal basis of T_{p}M. One can easily verify by simple linear algebra that the Ricci tensor is welldefined independently of the choice of orthonormal basis (E_{1}, E_{2}, …, E_{n}). Now

R(E_{i}, U, E_{i}, V) = R(E_{i}, V, E_{i}, U)

for all U and V, by Lemma 4.4. Thus ρ(U, V) = ρ(V, U) for all U, V ∈ T_{p}M. It can be shown that if V is a tangent vector of unit length at some point p of M then the avarage of the sectional curvatures in all planes in T_{p}M that contain the vector V is equal to

\frac{1}{n − 1}ρ(V, V),

where n is the dimension of n.

Theorem 8.8 (Myers) Let (M, g) be a Riemannian manifold of dimension n whose Ricci tensor ρ satisfies the condition ρ(V, V) ≥ C|V |^{2} for all tangent vectors V at all points of M. Let γ: [0, L] → M be a geodesic of length L in M parameterized by arclength. If L^{2} > (n − 1)π^{2}/C then the geodesic γ is not of minimal length. Thus if (M, g) is complete and connected then M is compact and the diameter of M does not exceed

\sqrt{\frac{(n − 1)π^{2}}{C}} .

Proof Let γ: [0, L] → M be a geodesic in M parameterized by arclength whose length L satisfies L^{2} > (n − 1)π^{2}/C. Choose parallel vector fields E_{1}, E_{2}, …, E_{n−1} along γ with the property that

(E_{1}(0), E_{2}(0), …, E_{n−1}(0), γ′(0))

is an orthonormal basis of T_{γ(0)}M. Then

(E_{1}(t), E_{2}(t), …, E_{n−1}(t), γ′(t))

is an orthonormal basis of T_{γ(t)}M for all t ∈ [0, L]. (This follows from Lemma 4.2 on account of the fact that the vector fields E_{1}, E_{1}, …, E_{n−1}, γ′ are covariantly constant along the geodesic γ.) Thus

ρ(γ′(t), γ′(t)) = ∑_{i=1}^{n−1} R(E_{i}(t), γ′(t), E_{i}(t), γ′(t))

(where we have used the fact that R(γ′(t), γ′(t), γ′(t), γ′(t)) = 0.) Let us define smooth vector fields V_{1}, V_{2}, …, V_{n−1} along γ by

V_{i}(t) = sin (\frac{πt}{L}) E_{i}(t).

Then V_{i}(0) = 0 and V_{i}(L) = 0 for all i. Then

∑_{i=1}^{n−1} R(V_{i}(t), γ′(t), V_{i}(t), γ′(t)) = sin^{2} (\frac{πt}{L}) ρ(γ′(t), γ′(t)) ≥ C sin^{2} (\frac{πt}{L}) .

But

g (V_{i}(t), \frac{D^{2}V_{i}(t)}{dt^{2}}) = −\frac{π^{2}}{L^{2}} sin^{2} (\frac{πt}{L}) .

Thus if L^{2} > (n − 1)π^{2}/C then

∑_{i=1}^{n−1} (g (V_{i}(t), \frac{D^{2}V_{i}(t)}{dt^{2}}) + R(V_{i}(t), γ′(t), V_{i}(t), γ′(t))) > 0

for all t ∈ (0, L). But if γ were a geodesic of minimal length then we would have

∑_{i=1}^{n−1} ∫_{a}^{b} (g (V_{i}(t), \frac{D^{2}V_{i}(t)}{∂t^{2}}) + R(V_{i}(t), γ′(t), V_{i}(t), γ′(t))) dt ≤ 0,

by Lemma 8.6. This shows that if the length L of γ satisfies L^{2} > (n−1)π^{2}/C then γ is not a geodesic of minimal length.

Now suppose that (M, g) is complete and connected. Then any two points of M can be joined by a geodesic of minimal length, by Theorem 7.1. We conclude that the length L of this geodesic must satisfy L^{2} > (n − 1)π^{2}/C, so that the diameter of M does not exceed

\sqrt{\frac{(n − 1)π^{2}}{C}} .

Moreover if we choose some p ∈ M then M is contained in the image under the exponential map exp_{p} of the closed ball in T_{p}M of radius R about the zero vector, where R^{2} = (n − 1)π^{2}/C. Thus M is compact (since M is the image of a compact set under a continuous map).

Remark Let S^{n} denote the unit sphere in ℝ^{n+1}. We define the standard Riemannian metric g on S^{n} to be the restriction to the tangent spaces of S^{n} of the standard inner product on ℝ^{n+1}. When one calculates the Riemann curvature tensor of S^{n} one finds that all the sectional curvatures of S^{n} are equal to +1, and the Ricci tensor ρ aof S^{n} is given by ρ(U, V) = (n−1)g(U, V) for all tangent vectors U and V at any given point of S^{n}. Any meridian of S^{n} joining the points (1, 0, …, 0) and (−1, 0, …, 0) is a geodesic of length π. It follows that the inequality L^{2} ≤ (n − 1)π^{2}/C on the length L of minimal geodesics provided by Myers’ Theorem is an equality for this geodesic on S^{n} (where C is given by C = n − 1). Thus the upper bound on the length of geodesics of minimal length provided by Myers’ Theorem is attained when the Riemannian manifold in question is the standard n-dimensional sphere S^{n}.

Remark It can be shown that if M is a connected smooth manifold then there exists a covering map π: M̃ → M over M, where M̃ is a simpyconnected smooth manifold. The smooth manifold M̃ is known as the universal cover of the manifold M. Moreover the covering map πM̃ → M is a local diffeomorphism. The inverse image π^{−1}(m) of a point m of M is in bijective correspondence with the fundamental group π_{1}(M, m) of M based at M. Any Riemannian metric g on M induces a corresponding Riemannian metric g̃ on M̃ characterized by the property that the covering map π: M̃ → M is a local isometry. The Riemannian manifold (M̃, g̃) is complete if and only if (M, g) is complete.

Now suppose that M is connected and complete and that there exists a positive constant C > 0 with the property that ρ(V, V) ≥ C|V |^{2} for all tangent vectors V on M, where ρ is the Ricci tensor of M. Then the same condition will hold for the universal cover (M̃, g̃) of M. We conclude that M̃ is compact. Now given m ∈ M, there exists an open neighbourhood U of M such that π^{−1}(U) is a disjoint union of copies of U. It follows from the compactness of M̃ that the number of such copies must be finite. Therefore the fundamental group of M is finite.

With substantially more effort, one can prove the following theorem.

Sphere Theorem Let (M, g) be a complete simply-connected Riemannian manifold of dimension n. Suppose that the sectional curvatures K(P) of M satisfy \frac{1}{4}K_{0} < K(P) ≤ K_{0} for some positive constant K_{0}. Then M is homeomorphic to the n-dimensional sphere S^{n}.