Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 41 (2000), No. 2, 359-370.

Immersions with a Parallel Normal Field

Sheila Carter Yusuf Kaya^{1}

School of Mathematics, University of Leeds, Leeds, LS2 9JT, England

Abstract. In this paper we investigate the restrictions which are placed on the focal set of a submanifold of Euclidean space under the assumption that the submanifold has a parallel normal field.

1. Introduction

Throughout this paper f : M^{m} → ℝ^{m+k} is a C^{∞} immersion of a connected C^{∞} m-dimensional manifold M without boundary into Euclidean (m + k)-space.

The tangent space of M at a point p will be denoted by T_{p}M. So df_{p} : T_{p}M → T_{f(p)}(ℝ^{m+k}) = {f(p)} × ℝ^{m+k} ≡ ℝ^{m+k} is an injection.

There is a standard inner product <, > on ℝ^{m+k}. The total space of the normal bundle of f is defined by

N(f) = {(p, v) ∈ M × ℝ^{m+k} : < v, df_{p}(X) >= 0 ∀X ∈ T_{p}M} ,

and the affine normal k-plane to f(M) at p is ν_{p}(f) = {f(p) + v : (p, v) ∈ N(f)}. Note that N(f) is an (m + k)-dimensional manifold. The endpoint map η(p, v) = f(p) + v. A point x ∈ ℝ^{m+k} is a focal point of M with base p if η is singular at (p, x − f(p)). The focal point has multiplicity μ > 0 if rank(Jac η) = m + k − μ at that point. The set of focal points of f with base p will be denoted by F_{p}(f). This is an algebraic variety in ν_{p}(f). We remark that by [9], x ∈ F_{p}(f) if and only if x ∈ ν_{p}(f), x = f(p) + \frac{1}{λ}ξ where ξ = \frac{x−f(p)}{∥x−f(p)∥}, and λ is an eigenvalue of the shape operator A_{ξ} : T_{p}M → T_{p}M, i.e. λ is a principal curvature of f at p in the normal direction ξ.

1The second author is supported by Zonguldak Karaelmas University, Turkey 0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2000 Heldermann Verlag

If f has flat normal bundle it is known that for each p ∈ M, F_{p}(f) is a union of at most m hyperplanes in the normal plane ν_{p}(f) [17], [10]. The origin of this fact can be traced back to [18]. By Corollary 1.5 in [14], if there is a unit parallel normal field ξ for f such that the shape operator A_{ξ} has m distinct eigenvalues, then the normal bundle of f is flat. Also for any k ≥ 2, if there exists locally an orthonormal set of (k − 1) parallel normal fields ξ_{1}, … ξ_{k−1} for f at each point of M, then f has flat normal bundle [17].

In this paper we consider the case where f has a parallel normal field and show that for k ≥ 2 and for all p ∈ M, each 2-plane in ν_{p}(f) which contains the normal line at f(p) in the direction of this parallel normal field intersects F_{p}(f) in the union of at most m straight lines. We use this to compare critical points of height and distance functions for f with those for parallel immersions to f.

2. Parallel immersions and focal sets

Let f : M^{m} → ℝ^{m+k} be an immersion and assume that there exists a parallel normal field ξ : M^{m} → ℝ^{m+k} for f. The map

f_{ξ} : M^{m} → ℝ^{m+k} is defined by f_{ξ}(p) = f(p) + ξ(p).

If f_{ξ} is an immersion, it is called a parallel immersion to f and ξ is said to be immersive. We state the following important relations between f and f_{ξ}. Formal proofs can be found in [11], [13].

Theorem 2.1. Let f : M^{m} → ℝ^{m+k} be an immersion and assume that there exists a parallel normal field for f, ξ : M^{m} → ℝ^{m+k}, then

(i) f_{ξ} is an immersion if and only if, for all p ∈ M, f_{ξ}(p) ∉ F_{p}(f), (ii) ν_{p}(f_{ξ}) = ν_{p}(f) for all p ∈ M, (iii) x ∈ ℝ^{m+k} is a focal point of f_{ξ} with base p if and only if x is a focal point of f with

base p, i.e. F_{p}(f_{ξ}) = F_{p}(f) for all p ∈ M.

When f_{ξ} is an immersion, the index of f_{ξ}, ind f_{ξ} is defined to be the total multiplicity of the focal points of f with base p on the line segment between f(p) and f_{ξ}(p). As observed in [16], [2], it follows from the above theorem that this integer is a constant for all p ∈ M. We also call this integer the index of ξ.

For a given immersion f : M → ℝ^{m+k} we may not have a parallel normal field on M, even locally. But we can compose f with an inclusion of ℝ^{m+k} in ℝ^{m+k+1} or with an inverse stereographic projection from ℝ^{m+k} to 𝕊^{m+k} ⊂ ℝ^{m+k+1} to get a parallel normal field, namely define

g : M → ℝ^{m+k+1} by g(p) = (f(p), 0) and
g̃ : M → 𝕊^{m+k}  by g̃(p) = (\frac{2f(p)}{∥f(p)∥^{2} + 1}, \frac{∥f(p)∥^{2} − 1}{∥f(p)∥^{2} + 1}) .

Then ξ = (0, 0, …, 0, 1) and ξ̃ = g̃ are parallel normal fields for g, g̃ respectively. Now, for p ∈ M, let Q ⊂ ν_{p}(g) and Q̃ ⊂ ν_{p}(g̃) be 2-planes containing g(p), g_{ξ}(p) and g̃(p), g̃ξ̃(p) respectively. Then one can calculate that Q ∩ F_{p}(g) and Q̃ ∩ F_{p}(g̃) are either empty or a union of at most m lines; in fact Q ∩ F_{p}(g) is either empty or a union of at most m parallel lines and Q̃ ∩ F_{p}(g̃) can only be the union of m lines (counting multiplicities) intersecting at the centre of the unit sphere 𝕊^{m+k}.

The aim of this section is to prove the following theorem which is motivated by the above two examples. When there can be no confusion we abbreviate ν_{p}(f), F_{p}(f) to ν_{p}, F_{p}.

Theorem 2.2. Let f : M^{m} → ℝ^{m+k} be an immersion with k ≥ 2 and assume that there exists a parallel normal field for f, ξ : M^{m} → ℝ^{m+k}. For p ∈ M, let Q ⊂ ν_{p} be a 2-subplane containing f(p) and f_{ξ}(p). Then either Q ∩ F_{p} = ∅ or Q ∩ F_{p} is the union of at most m straight lines.

Note that this theorem could be proved from the fact that the shape operator A_{ξ} commutes with the shape operator for any other normal direction. A more sophisticated method is to use a partial tube as defined in [3] with type fibre a (k − 2)-sphere with a line as axis. The axis is defined by the given parallel normal field and the methods of [4] show that this partial tube has flat normal bundle. The fibres of this normal bundle can be identified with the 2-planes referred to in the theorem and the lines in these planes together with the axes of the spheres are the focal sets of the partial tube. For more details on this method and a generalisation of Theorem 2.2 see [1]. However here we prefer to use a more direct geometrical approach which throws light on the structure of focal sets, and for this we need the following two propositions.

Proposition 2.3. Let f : M^{m} → ℝ^{m+k} be an immersion, let p ∈ M and let Q be an nsubplane of ν_{p} through f(p). Then Q ∩ F_{p} is isometric to the zero set in ℝ^{n} of a polynomial h ∈ ℝ[t_{1}, …, t_{n}] of degree at most m.

Proof. Let p ∈ M, then without loss of generality we can assume that f(p) is at the origin in ℝ^{m+k}. Let {e_{1}, e_{2}, …, e_{n}} be an orthonormal basis of Q. Then y ∈ Q is a focal point of f with base p if and only if

y = ∑_{i=1}^{n} t_{i}e_{i} and det(I − ∑_{i=1}^{n} t_{i}A_{i}) = 0

where A_{i} : T_{p}M → T_{p}M is the shape operator of f at p in the normal direction e_{i} . So Q∩F_{p} is isometric to the zero set of the polynomial h(t_{1}, …, t_{n}) = det(I − ∑_{i=1}^{n} t_{i}A_{i}).

P. J. Ryan proved that given a codimension one immersion of a manifold M with unit parallel normal field ξ, there exist continuous functions {λ_{i} : M → ℝ}_{1≤i≤m} such that each λ_{i}(p) is an eigenvalue of the shape operator A_{ξ(p)} [12]. By a similar technique we prove that in the higher codimension case, for each p ∈ M, the focal points of f with base p on normal lines through f(p) vary continuously with respect to the normal direction.

Proposition 2.4. Let f : M^{m} → ℝ^{m+k} be an immersion, let p ∈ M and let Q be a 2subplane of ν_{p} through f(p) with orthonormal basis {e_{1}, e_{2}}. Then there exist continuous functions {λ_{i} : 𝕊^{1} → ℝ}_{1≤i≤m} such that, for all θ ∈ 𝕊^{1}, λ_{1}(θ), …, λ_{m}(θ) are the eigenvalues of the shape operator A_{η(θ)} at p in the direction η(θ) = cos θe_{1} + sin θe_{2}, and λ_{1}(θ) ≥ λ_{2}(θ) ≥ ⋯ ≥ λ_{m}(θ).

Proof. Let g(t, θ) = t^{m} + a_{1}(θ)t^{m−1} + ⋯ + a_{m}(θ) be the characteristic polynomial of A_{η(θ)}. Consider g(z, θ) as a complex polynomial in z.

For θ_{0} ∈ 𝕊^{1}, let {μ_{i}(θ_{0})}_{1≤i≤r} be the distinct eigenvalues of A_{η(θ_{0})} and let {m_{i}}_{1≤i≤r} be their respective multiplicities. So ∑_{i=1}^{r} m_{i} = m. Assume μ_{1}(θ_{0}) > μ_{2}(θ_{0}) > ⋯ > μ_{r}(θ_{0}) .

Let ϵ > 0 be arbitrary, let ϵ_{0} = min \{ϵ, \frac{1}{2}, \frac{1}{2} min_{\begin{matrix}1≤i,j≤r \\ i≠j \end{matrix}}|μ_{i}(θ_{0}) − μ_{j}(θ_{0})|\} and let C_{i} = \{z ∈ ℂ :

|z − μ_{i}(θ_{0})| = ϵ_{0}}.

Clearly g(z, θ_{0}) ≠ 0 on C_{i}. Now choose δ_{0} > 0 so that if |θ − θ_{0}| < δ_{0} then g(z, θ) ≠ 0 on C_{i}. Now

m_{i} = \frac{1}{2πi} ∫_{C_{i}} \frac{g′(z, θ_{0})}{g(z, θ_{0})}dz,

and

|\frac{1}{2πi} ∫_{C_{i}} (\frac{g′(z, θ_{0})}{g(z, θ_{0})} − \frac{g′(z, θ)}{g(z, θ)})dz| ≤ ϵ_{0} sup_{z∈C_{i}} |\frac{g′(z, θ_{0})}{g(z, θ_{0})} − \frac{g′(z, θ)}{g(z, θ)}|.

When θ → θ_{0}, the right hand side converges uniformly to 0 on C_{i}, thus there exists δ < δ_{0} such that |θ − θ_{0}| < δ implies

sup_{z∈C_{i}} |\frac{g′(z, θ_{0})}{g(z, θ_{0})} − \frac{g′(z, θ)}{g(z, θ)}| < 1.

Hence, as the integral \frac{1}{2πi} ∫_{C_{i}} \frac{g′(z,θ)}{g(z,θ)}dz is the number of zeros of g(z, θ) inside C_{i}, it is integer valued and is equal to m_{i}.

So |θ−θ_{0}| < δ implies that g(z, θ) has m_{i} roots inside C_{i}. For each θ ∈ 𝕊^{1}, let {λ_{i}(θ)}_{1≤i≤m} be the m eigenvalues of A_{η(θ)} with λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{m}. They are real as A_{η(θ)} is real symmetric, and {λ_{1}(θ_{0}), λ_{2}(θ_{0}), ⋯, λ_{m}(θ_{0})} = {μ_{1}(θ_{0}), μ_{2}(θ_{0}), ⋯, μ_{r}(θ_{0})}.

So if |θ − θ_{0}| < δ then |λ_{i}(θ) − λ_{i}(θ_{0})| < ϵ_{0} ≤ ϵ. So each λ_{i} is continuous.

Proof of Theorem 2.2. By Proposition 2.3 we can identify Q with ℝ^{2}, f(p) with (0, 0) and F_{p}^{Q} = F_{p} ∩ Q with {(x, y) ∈ ℝ^{2} : h(x, y) = 0} where h is a polynomial in 2 variables of degree ≤ m. By the Unique Factorization Theorem [8] we can write h as

h(x, y) = c ∏_{i=1}^{r} h_{i}(x, y)^{m_{i}}

where c is a real number, each h_{i} is an irreducible polynomial in ℝ[x, y], for i ≠ j h_{i} is not a

factor of h_{j} and m_{1} + m_{2} + ⋯ + m_{r} ≤ m. Then F_{p}^{Q} is the zero set of ∏^{r} ; i=1h_{i}(x, y). We know

that for i ≠ j, h_{i} and h_{j} have finitely many intersection points and also each h_{i} has finitely many singular points [7]. Let W denote the complement in F_{p}^{Q} of this finite set of points. Assume that the factors of h are ordered so that W ∩ h_{i}^{−1}(0) ≠ ∅ if and only if i = 1, …, s.

Then the zero set of ∏^{r} ; i=s+1h_{i}(x, y) is a finite set of points, P say, and for i = 1, …, s the zero

set of each h_{i} is a piecewise smooth curve with finitely many components.

Fix i ∈ {1, …, s} and let (a, b) ∈ W be such that h_{i}(a, b) = 0 and (a, b) is a regular point of h_{i}. Then, without loss of generality, we can assume \frac{∂h_{i}(a,b)}{∂y} ≠ 0, and therefore by the Implicit Function Theorem we get a smooth function β such that h_{i}(x, β(x)) = 0 and (x, β(x)) ∈ W for all x ∈ (a − ϵ, a + ϵ) for some ϵ > 0. Define the smooth curve α by α(x) = (x, β(x)). We will show that the curvature of α, κ(x), is zero for all x ∈ (a − ϵ, a + ϵ).

Let ℓ denote the line {f(p) + tξ(p) : t ∈ ℝ} and for x ∈ (a − ϵ, a + ϵ) let ℓ_{x} denote the tangent line to α(a − ϵ, a + ϵ) at α(x). PSfrag replacements

ℓ_{u}

ℓ
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α(u−γ, u+γ)

f_{sξ}(p)

Figure 1.

Suppose κ_{α}(x_{0}) ≠ 0 for some x_{0} ∈ (a − ϵ, a + ϵ), then there exists δ > 0 such that κ(x) ≠ 0 for all x ∈ (x_{0} − δ, x_{0} + δ) ⊂ (a − ϵ, a + ϵ). Then an elementary calculation shows that {ℓ ∩ ℓ_{x} : x ∈ (x_{0} − δ, x_{0} + δ)} contains an open interval. So, as there are only a finite number of focal points with base p on ℓ there exits u ∈ (x_{0} − δ, x_{0} + δ) such that ℓ ∩ ℓ_{u} ≠ ∅ and ℓ ∩ ℓ_{u} ∉ F_{p}^{Q}.

Let ℓ ∩ ℓ_{u} = f(p) + sξ(p) = f_{sξ}(p). Since f_{sξ}(p) ∉ F_{p} we can use Theorem 2.1 to show that there exists a neighbourhood U of p in M such that f_{sξ}|U is an immersion and has the same focal set as f|U. Thus F_{p}^{Q} is the focal set of f_{sξ}|U at p.

Now as κ(u) ≠ 0 there exists γ > 0 such that α(u − γ, u + γ) lies on one side of the tangent line ℓ_{u}, see Figure 1. But, as ℓ_{u} ⊂ Q, it is a normal line to f_{sξ} at p. So α(u) = f_{sξ}(p)+\frac{1}{λ(θ)}η(θ) for some principal curvature function λ of f_{sξ} as defined in Proposition

2.4, where η(θ) = \frac{α(u)−f_{sξ}(p)}{∥α(u)−f_{sξ}(p)∥} determines the unit direction of ℓ_{u}. But λ is not continuous at θ since the focal points of f_{sξ} at p do not change continuously as the normal direction changes continuously (there is no focal point near α(u) on normal lines moving in one direction away from ℓ_{u}). This contradicts Proposition 2.4 for f_{sξ}.

Hence κ(x) = 0 for all x ∈ (a − ϵ, a + ϵ) and therefore α(a − ϵ, a + ϵ) is a straight line segment. Since this is in the zero set of h_{i} and h_{i} is irreducible it follows that the zero set of h_{i} is a straight line, Λ_{i} say, and degree of h_{i} = 1.

So the focal set F_{p}^{Q} consists of the focal lines Λ_{1}, …, Λ_{s} with corresponding multiplicities m_{1}, …, m_{s}, together with the finite set of points P. Now we are going to show that P = ∅.

For θ ∈ ℝ let ℓ_{θ} denote the line {f(p) + t(cos θe_{1} + sin θe_{2}) : t ∈ ℝ}, where e_{1}, e_{2} is an orthonormal basis of Q and let

Θ = {θ ∈ ℝ : ℓ_{θ} ∩ P = ∅ and ℓ_{θ} ∩ Λ_{i} ≠ ∅, ∀i = 1, …, s}.

Then ℝ\Θ is finite and ∀θ ∈ Θ the total multiplicity of focal points with base p on ℓ_{θ} is ∑^{s} ; i=1m_{i}.

Hence there is a “focal point at infinity on ℓ_{θ}” with multiplicity m − ∑^{s} ; i=1m_{i}, i.e. the shape

operator of f at p in direction η(θ) = cos θe_{1} + sin θe_{2} has an eigenvalue 0 with multiplicity

m − ∑^{s} ; i=1m_{i}. It follows from Proposition 2.4 that the line at infinity in Q (extending Q to a

projective plane) is a focal line with multiplicity m− ∑^{s} ; i=1m_{i}, and ∀θ ∈ ℝ, the total multiplicity

of the eigenvalues of A_{η(θ)} at p arising from the focal lines is m. (Note if ℓ_{θ} ∩ Λ_{i} = ∅ then the multiplicity m_{i} transfers to the eigenvalue 0). Hence P = ∅. Thus if F_{p}^{Q} ≠ ∅ it is a union of straight lines.

Note that we need only the existence of a parallel normal field in a neighbourhood of p to obtain this result.

3. Parallel immersions, distance functions and height functions

In this section we obtain relations between the critical points of distance or height fuctions for f and those for f_{ξ}. As an application we prove the following theorem, which generalises Theorem 3.6(i) from [2].

Theorem 3.1. Let f : M^{m} → ℝ^{m+k} be an immersion of a compact manifold, and suppose that there exists an immersive parallel normal field for f with odd index. Then the Euler characteristic χ(M) = 0.

Definition 3.2. Let f : M^{m} → ℝ^{m+k} be an immersion, let ξ : M^{m} → ℝ^{m+k} be an immersive parallel normal field for f and let x ∈ ℝ^{m+k}. The distance functions L_{x} : M^{m} → ℝ, L̂_{x} : M^{m} → ℝ for f, f_{ξ} respectively are defined by L_{x}(p) = ∥x − f(p)∥^{2}, L̂_{x}(p) = ∥x − f_{ξ}(p)∥^{2} for all p ∈ M.

Theorem 3.3. Let f : M^{m} → ℝ^{m+k} be an immersion, let ξ : M^{m} → ℝ^{m+k} be an immersive parallel normal field for f with index λ and let x ∈ ℝ^{m+k}. Then p is a (nondegenerate)critical point of L_{x} if and only if p is a (nondegenerate)critical point of L̂_{x}. Further, if p is a nondegenerate critical point of L_{x} with index μ then ∃ ℓ ∈ ℕ, max{0, λ + μ − m} ≤ ℓ ≤ min{λ, μ} such that p is a nondegenerate critical point of L̂_{x} with index λ + μ − 2ℓ.

Proof. We know that p is a critical point of L_{x} if and only if x ∈ ν_{p}(f), and further p is nondegenerate if and only if x ∉ F_{p}(f) [9]. The proposition then follows immediately from Theorem 2.1 which states that ν_{p}(f) = ν_{p}(f_{ξ}) and F_{p}(f) = F_{p}(f_{ξ}).

So now assume that p is a nondegenerate critical point of L_{x} with index μ. We will write ac^{⌢} for the open line segment between points a and c in ν_{p}(f) and #ac^{⌢} for the total multiplicity of focal points with base p on ac^{⌢}.

To simplify the notation we will write a = f(p) and c = f_{ξ}(p). Now x ∈ ν_{p}(f) and #xa^{⌢} = μ [9]. Also, since ind f_{ξ} is λ we have #ac^{⌢} = λ.

If x is on the line through f(p) and f_{ξ}(p), then the index of p as a nondegenerate critical point of L̂_{x} is

#xc^{⌢} = \{ \begin{matrix}λ + μ & if x ∈ \{f(p) + tξ(p) : t < 0\} \\ λ − μ & if x ∈ \{f(p) + tξ(p) : 0 ≤ t ≤ 1\} \\ μ − λ & if x ∈ \{f(p) + tξ(p) : t > 1\} \end{matrix}

which corresponds to l = 0, μ or λ respectively.

Otherwise, if x is not on this line, consider the triangle, △, with vertices x, f(p),f_{ξ}(p) in ν_{p} as in Figure 2, and let Q be the 2-subplane of ν_{p} which contains △.

PSfrag replacements a=f(p)

c=f_{ξ}(p)
x
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λ−ℓ
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Q

Figure 2.

We know by Theorem 2.2, that Q ∩ F_{p} is a union of at most m lines counting multiplicities. Since none of the vertices of △ are in F_{p} it follows that none of the edges of △ can be part of a focal line, and therefore any focal line which intersects one edge of the triangle must intersect another edge.

Let ℓ be the total multiplicity of focal lines which intersect both xa^{⌢} and ac^{⌢}. So 0 ≤ ℓ ≤ min{λ, μ}. Then μ − ℓ lines intersect both xa^{⌢} and xc^{⌢}, and λ − ℓ lines intersect both ac^{⌢} and xc^{⌢} . Therefore #xc^{⌢} = (μ − ℓ) + (λ − ℓ) = λ + μ − 2ℓ ≤ m. Hence the index of p as a critical point of L̂_{x} is λ + μ − 2ℓ. Also the total multiplicity of focal lines which intersect △ is ℓ+(λ−ℓ)+(μ−ℓ) = (λ+μ−ℓ) ≤ m. Hence (λ + μ − m) ≤ ℓ.

We next prove corresponding results for height functions.

Definition 3.4. Let f : M^{m} → ℝ^{m+k} be an immersion, let ξ : M^{m} → ℝ^{m+k} be an immersive parallel normal field for f and let z ∈ 𝕊^{m+k−1}. The height functions H_{z} : M^{m} → ℝ, Ĥ_{z} :

M^{m} → ℝ for f, f_{ξ} respectively are defined by H_{z}(p) =< z, f(p) >, Ĥ_{z}(p) =< z, f_{ξ}(p) > for all p ∈ M.

Theorem 3.5. Let f : M^{m} → ℝ^{m+k} be an immersion, let ξ : M^{m} → ℝ^{m+k} be an immersive parallel normal field for f with index λ and let z ∈ 𝕊^{m+k−1}. Then p is a (nondegenerate) critical point of H_{z} if and only if p is a (nondegenerate) critical point of Ĥ_{z}. Further, if p is a nondegenerate critical point of H_{z} with index μ then ∃ ℓ ∈ ℕ with max{0, λ + μ − m} ≤ ℓ ≤ min{λ, μ} such that p is a nondegenerate critical point of Ĥ_{z} with index λ + μ − 2ℓ.

Proof. It is known that p is a critical point of H_{z} if and only if f(p) + z ∈ ν_{p}(f) [6]. This holds if and only if f(p) + ξ(p) + z = f_{ξ}(p) + z ∈ ν_{p}(f) = ν_{p}(f_{ξ}), that is, if and only if p is a critical point of Ĥ_{z}.

Now assume that p is a nondegenerate critical point of H_{z} with index μ. Let f^{−}(p) = {f(p) + tz : t < 0} and f^{+}(p) = {f(p) + tz : t > 0} with corresponding definitions for f_{ξ}^{−}(p) and f_{ξ}^{+}(p). Then, with the notation used in Theorem 3.3, f(p) + z ∈ ν_{p}(f), #f^{−}(p) = μ and #f^{+}(p) = m − μ. Also we have #ac^{⌢} = λ where a = f(p) and c = f_{ξ}(p).

If ξ(p) = tz for some t ∈ ℝ, then #f_{ξ}^{−}(p) + #f_{ξ}^{+}(p) = #f^{−}(p) + #f^{+}(p) = m, since F_{p}(f_{ξ}) = F_{p}(f) and the line through f_{ξ}(p) and f_{ξ}(p) + z coincides with the line through f(p) and f(p) + z. Hence p is a nondegenerate critical point of Ĥ_{z} and its index is

#f_{ξ}^{−}(p) = \{ \begin{matrix}λ + μ & if t ≥ 0 \\ μ − λ & if t < 0 \end{matrix} ,

which correponds to ℓ = 0, λ in the statement of the theorem.

Otherwise, if ξ(p) ≠ tz for all t ∈ ℝ, let Q be the 2-subplane of ν_{p} which contains f(p), f(p) + z and f_{ξ}(p) as in Figure 3.

Since #(f^{−}(p) ∪ f^{+}(p)) = m it follows from Theorem 2.2 that Q ∩ F_{p} is a union of m lines counting multiplicity and each line intersects the line through f(p) and f(p) + z transversally and hence intersects the line through f_{ξ}(p) and f_{ξ}(p)+z transversally. Therefore #(f_{ξ}^{−}(p) ∪ f_{ξ}^{+}(p)) = m and p is a nondegenerate critical point of Ĥ_{z}.

Let ℓ be the total multiplicity of focal lines which intersect both f^{−}(p) and ac^{⌢}. So 0 ≤ ℓ ≤ min{λ, μ}. Then these lines intersect f_{ξ}^{+}(p). The remaining μ − λ lines which intersect f^{−}(p) also intersect f_{ξ}^{−}(p). Since λ lines must intersect ac^{⌢} it follows that λ − ℓ lines intersect f^{+}(p) and f_{ξ}^{−}(p). So 0 ≤ λ − ℓ ≤ m − μ, which gives ℓ ≥ λ + μ − m.

We then get that #f_{ξ}^{−}(p) = (μ − ℓ) + (λ − ℓ) = λ + μ − 2ℓ. Hence the index of p as a critical point of Ĥ_{z} is λ + μ − 2ℓ.

If we start with a nondegenerate critical point p of Ĥ_{z}, then applying the above argument to f_{ξ} and (f_{ξ})_{−ξ} = f shows that p is a nondegenerate critical point of H_{z}.

Proof of Theorem 3.1. We are now assuming that M is compact. There exists z ∈ 𝕊^{m+k−1} such that H_{z} is a nondegenerate height function for f. Then by Theorem 3.5, Ĥ_{z} is a nondegenerate height function for f_{ξ}. Let C_{μ} (resp. D_{μ}) denote the number of critical points of H_{z} (resp. Ĥ_{z}) with index μ. By Theorem 3.5, since λ is odd, p is an even (resp. odd) indexed critical point of H_{z} if and only if p is an odd (resp. even) indexed critical point of Ĥ_{z}.

PSfrag replacements

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z
f(p)
f_{ξ}(p)
μ
λ
ℓ
λ−ℓ
m−μ

Figure 3.

Therefore

∑_{μ even} C_{μ} = ∑_{μ odd} D_{μ} and ∑_{μ odd} C_{μ} = ∑_{μ even} D_{μ} .

Now, by the Weak Morse inequalities [9],

χ(M) = ∑_{μ=0}^{m}(−1)^{μ}C_{μ} = ∑_{μ even} C_{μ} − ∑_{μ odd} C_{μ} = ∑_{μ even} D_{μ} − ∑_{μ odd} D_{μ}
= ∑_{μ odd} C_{μ} − ∑_{μ even} C_{μ} (from the above relations)
= −χ(M) .

Hence χ(M) = 0.

4. Parallel immersions, tightness and tautness

Finally, in this section we draw some conclusions from the hypothesis that a taut or tight immersion has an immersive parallel normal field.

Proposition 4.1. Let f : M^{m} ⟶ ℝ^{m+k} be an immersion of a compact manifold and suppose f has an immersive parallel normal field with index λ. Then, ∀μ ∈ {0, λ, m − λ, m}, any nondegenerate distance or height function for f has a critical point with index μ.

Proof. Let H_{z} be a nondegenerate height function for f, for some z ∈ 𝕊^{m+k−1}. Since M is compact, H_{z} and Ĥ_{z} have critical points with indices 0 and m. By Theorem 3.5 applied to f_{ξ} and (f_{ξ})_{−ξ} = f, if p is a critical point of Ĥ_{z} with index 0 (resp. m) then p is a critical point of H_{z} with index λ corresponding to ℓ = 0 (resp. m − λ corresponding to ℓ = λ).

Likewise using Theorem 3.3 a similar argument applies to distance functions.

Corollary 4.2. Let f : M^{m} ⟶ ℝ^{m+k} be an immersion of a compact manifold and let 0 < λ < m. If there exists a nondegenerate distance (resp. height) function for f which has no critical point with index λ then there does not exist an immersive parallel normal field for f with index λ nor with index m − λ.

Definition 4.3. The Morse number γ(M) of a manifold M is defined by

γ(M) = inf { ∑_{λ=0}^{m} C_{λ}(α) : α is a nondegenerate function on M}.

Then an immersion f : M^{m} ⟶ ℝ^{m+k} is taut (resp. tight) if every nondegenerate distance (resp. height) function has γ(M) critical points.

In particular if the number of critical points of every distance (resp. height) function is equal to the sum of Betti numbers of M with respect to some field then f is taut (resp. tight) by using the Weak Morse Inequalities [9].

Proposition 4.4. Let f : M^{m} ⟶ ℝ^{m+k} be an immersion. Suppose that there exists an immersive parallel normal field for f, ξ : M^{m} ⟶ ℝ^{m+k}. Then f is taut (resp. tight) if and only if f_{ξ} is taut (resp. tight).

Proof. This is immediate from the Theorems 3.3 and 3.5 since L_{x} and L̂_{x} (resp. H_{z} and Ĥ_{z}) have the same number of critical points.

Proposition 4.5. Let f : M^{m} ⟶ ℝ^{m+k} be a tight or taut immersion of a compact manifold such that f has an immersive parallel normal field with index λ for some 0 < λ < m. Then γ(M) ≥ 4.

Proof. This is a straightforward application of Proposition 4.1. Note that if λ = m − λ then H_{z} must have 2 critical points with index λ corresponding to an index 0 and an index m critical point of Ĥ_{z}. Similarly for L_{x}.

Finally we consider a 2m-dimensional, compact, (m − 1)-connected manifold M. The diffeomorphism types of such taut submanifolds are determined in [15] . Here we use the methods of [5].

Theorem 4.6. Let f : M^{2m} ⟶ ℝ^{2m+k} be a substantial taut embedding of an (m − 1)- connected, compact, 2m-dimensional manifold such that f has a parallel normal field ξ then either k = 1 or f(M) ⊂ 𝕊^{2m+k−1}.

Proof. Suppose that k > 1 and let p ∈ M. Since f is substantial every normal line through f(p) meets the focal set F_{p} [5]. Further as each distance function for f can only have nondegenerate critical points with index 0, m or 2m it follows that on each normal line through f(p) there can be either one focal point with multiplicity m or 2m, or two focal points each with multiplicity m. Now let Q ⊂ ν_{p} be a 2-subplane which contains f(p) and f_{ξ}(p). Then by Theorem 2.2, Q ∩ F_{p} ⊂ ν_{p} is a union of lines, and by the above comments it must be two intersecting lines each with multiplicity m. Hence the point of intersection of the two lines is an umbilic in F_{p}. Thus f is spherical [5].

Theorem 4.7. Let f : M^{2m} ⟶ ℝ^{2m+k} be a substantial taut embedding of an (m − 1)- connected (not m-connected) compact manifold such that f has flat normal bundle then k ≤ 2 and if k = 1, f has an immersive parallel normal field with index m.

Proof. As f has flat normal bundle it follows that for all p in M, F_{p} is the union of at most 2m hyperplanes in ν_{p} [10].

For k ≥ 2, F_{p} must consist of 2 intersecting hyperplanes in ν_{p} each with multiplicity m otherwise, as in the proof of Theorem 4.6, f would not be substantial or there would exist a distance function for f which has a nondegenerate critical point with index not equal to 0, m or 2m. Further if k > 2 there is a line l in the intersection of the two hyperplanes and then the line l′ in ν_{p} through f(p) and parallel to the line l has no focal point on it. This contradicts f is substantial. Hence k ≤ 2.

Now assume k = 1 and let n : M^{2m} ⟶ ℝ^{2m+1} be the inward pointing unit normal. For each p ∈ M, consider the inward pointing normal ray ν_{p}^{+} = {f(p) + tn(p) : t > 0}. As in the proof of Theorem 3.8 of [5], F_{p} ∩ ν_{p}^{+} ≠ ∅ and consists of at most two points each with multiplicity m. Let f(p) + t_{1}(p)n(p) be the first point of F_{p} ∩ ν_{p}^{+}, and, if it exists, let f(p) + t_{2}(p)n(p) be the second such point, so t_{1}(p) < t_{2}(p). Put

d = sup{t_{1}(p) : p ∈ M}
e = inf{t_{2}(p) : p ∈ M} .

It is shown in [5] that 0<d<e <∞. Now choose c∈ℝ such that d<c<e, then for all p∈M, f(p) + cn(p) ∉ F_{p}. Hence ξ = cn is an immersive parallel normal field with index m.

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Received July 8, 1999