j. differential geometry 57 (2001) 233-238

THE METRIC FIBRATIONS OF EUCLIDEAN SPACE

DETLEF GROMOLL & GERARD WALSCHAP

Abstract

The purpose of this note is to complete the classification of metric fibrations in Euclidean space begun in [1]. Building on our techniques there, we show that regardless of dimension, the fibers are always the orbits of a free isometric group action by generalized glide rotations. A key ingredient of the argument is the fact that in the global setting, these fibrations satisfy a strong algebraic rigidity.

1. The fiber over a soul and the main result

We begin by recalling some general facts concerning metric fibrations π : ℝ^{n+k} → M^{n} that were established in [1]. Notationwise, X, Y, Z will denote local horizontal fields, T, U, V vertical ones, and lower-case letters refer to individual vectors. We write e = e^{h} + e^{v} ∈ H ⊕ V for the decomposition of e ∈ T ℝ^{n+k} into its horizontal and vertical parts. Thus, the integrability tensor A and the second fundamental tensor S are given by

A_{X}Y = \frac{1}{2}[X, Y]^{v} = ∇^{v}_{X}Y, S_{X}U = −∇^{v}_{U}X.

M has nonnegative sectional curvature by O’Neill’s formula, and is diffeomorphic to ℝ^{n} since the fibers of the fibration are connected. In particular, any soul of M consists of a single point. The fiber F over a soul is a totally geodesic affine subspace of Euclidean space, and up to congruence, F = ℝ^{k} × 0 ⊂ ℝ^{k} × ℝ^{n}.

The normal bundle ν of F has two Riemannian connections relevant to the present situation: One is the usual connection ∇^{h}, which is just the horizontal component of the Euclidean one. The other is the Bott

Received February 4, 2000. The first author was supported in part by the N.S.F., and the second by the O.U. Research Council.

connection ∇^{B} for which the basic fields along F are parallel sections of

ν. The connection difference form Ω = ∇^{h} − ∇^{B} is then the 1-form on F with values in the skew-symmetric endomorphism bundle of ν given by

Ω(U)X = −A_{X}^{∗}U,

where A_{X}^{∗} denotes the pointwise adjoint of A_{X}. When X, Y are basic, one always has

(1.1)  d(A_{X}Y)^{♭}(U, V) = ⟨dΩ(U, V)X, Y ⟩

for the 1-form (A_{X}Y)^{♭} metrically dual to A_{X}Y.

Our goal is to establish that Ω is Bott-closed, or equivalently, that each integrability field A_{X}Y is parallel on F for basic X, Y. The following main result is then an immediate consequence of [1, Theorem 2.6]:

Theorem. Let π : ℝ^{n+k} → M^{n} be a metric fibration of Euclidean space with connected fibers. Then

1. The fiber F over a soul of M is an affine subspace of Euclidean space, which, up to congruence, may be taken to be F = ℝ^{k} × 0.

2. The connection difference form Ω along the normal bundle of F induces a Lie algebra homomorphism Ω : ℝ^{k} → 𝔰𝔬(n), and π is the orbit fibration of the free isometric group action ψ of ℝ^{k} on ℝ^{n+k} = ℝ^{k} × ℝ^{n} given by

ψ(v)(u, x) = (u + v, ϕ(v)x), u, v ∈ ℝ^{k}, x ∈ ℝ^{n},

where ϕ : ℝ^{k} → SO(n) is the representation of ℝ^{k} induced by Ω.

2. Polynomial growth of the holonomy form

The mean curvature form of the fibration is the horizontal 1-form κ on ℝ^{n+k} given by κ(E) = tr S_{E^{h}}. By [1, Corollary 2.3], every metric fibration of Euclidean space is taut; i.e., κ is basic and exact. Let f denote the function on ℝ^{n+k} that vanishes on F and satisfies df = κ (observe that f is constant along fibers since κ is basic), and set V = e^{−f}. Define the holonomy form ω to be the k-form ω := V τ, where τ is the vertical volume form of the fibers of π; i.e., τ is the k-form on ℝ^{n+k} whose metric dual at a point p is given by

τ^{♯}(p) = u_{1} ∧ ⋯ ∧ u_{k},

where u_{1}, …, u_{k} denotes any oriented orthonormal basis of the tangent space to the fiber at p. It is well known that in general, the Lie derivative of τ in horizontal directions X satisfies

(2.1)  L_{X}τ = −κ(X) τ

vertically. Now let E_{1}, ⋯, E_{k} be an oriented orthonormal basis of parallel vector fields on F, and extend them smoothly to all of ℝ^{n+k} by setting

U_{i}(a, y) := E_{i}(a, 0) − A_{y}^{∗}E_{i}(a, 0), (a, y) ∈ ℝ^{k} × ℝ^{n}.

More precisely, U_{i}(a, y) = ∥ [E_{i}(a, 0) − A_{I_{(a,0)}y}^{∗}E_{i}(a, 0)], with ∥ denoting parallel translation from (a, 0) to (a, y), and I_{(a,y)} the canonical isomorphism of ℝ^{n+k} with its tangent space at (a, y). In order to avoid cumbersome notation, we shall from now on just assume these identifications. Observe that for horizontal lines γ originating at F, U_{i} ∘ γ is the holonomy Jacobi field along γ which equals E_{i} at γ(0), see [1].

Lemma 2.2. ω^{♯} = U_{1} ∧ ⋯ ∧ U_{k}.

Proof. We must show that V = τ(U_{1}, …, U_{k}). Both functions are by definition constant equal to 1 on F. Next, observe that that if X is the tangent field of a horizontal geodesic from F, then XV = −V κ(X), whereas

\begin{aligned}X(τ(U_{1}, …, U_{k})) & = L_{X}(τ(U_{1}, …, U_{k})) = (L_{X}τ)(U_{1}, …, U_{k}) \\ & = −τ(U_{1}, …, U_{k})κ(X) \end{aligned}

by (2.1). Here we have used the fact that L_{X}U_{i} = 0. The lemma clearly follows. q.e.d.

Lemma 2.3. The form U_{1} ∧⋯∧U_{k} is polynomial of degree at most k on every horizontal affine subspace.

Proof. Notice that the holonomy fields U_{i} are a priori linear only along each affine subspace a × ℝ^{n} orthogonal to F. It will later become apparent that they are in fact global Killing fields generating the isometric group action.

Let p ∈ ℝ^{n+k}, and q a point on the horizontal space H through p. By Lemma 2.2, ∧_{i}U_{i} is holonomy invariant, so that

∧_{i}U_{i}(q) = ∧_{i}[U_{i}(p) − (A_{q−p}^{∗} + S_{q−p})U_{i}(p)].

Thus, by translating the origin to p, it suffices to show that the map x ↦ ∧_{i}(E_{i} − A_{x}^{∗}E_{i} − S_{x}E_{i}) is polynomial of degree at most k in x. But this follows from the fact that x ↦ A_{x}^{∗}E + S_{x}E is a linear map. q.e.d.

Lemma 2.4. For any (a, 0) and (0, x) in ℝ^{k} × ℝ^{n}, U_{1} ∧ ⋯ ∧ U_{k} is polynomial in x on every affine line through (a, 0) in directions of the image of A_{x}.

Proof. We show that if f is a component of ∧_{i}U_{i}, then all derivatives of f of sufficiently high order vanish in directions A_{x}y. The result then follows from Taylor’s expansion. Notice that it is actually sufficient to establish this for directions (A_{x}y, y) (since the derivatives of order > k in directions (0, y) vanish by Lemma 2.3). Using Lemma 2.3 once more, it remains to show that both (a, 0) and (a+A_{x}y, y) belong to a common horizontal affine subspace. We claim, in fact, that they both belong to the horizontal space through (a, x): Clearly, (a, 0) does; as to the other point, just observe that (a+A_{x}y, y)−(a, x) = (A_{x}y, y −x) is orthogonal to the vertical space at (a, x), since

⟨(A_{x}y, y − x), (u, −A_{x}^{∗}u)⟩ = ⟨A_{x}y, u⟩ − ⟨y − x, A_{x}^{∗}u⟩ = 0.

q.e.d.

3. Constancy of integrability fields

In this section, we use the polynomial behavior of the holonomy form to deduce that each integrability field A_{X}Y is parallel along the totally geodesic fiber F. Before getting into the details of the argument, we provide a brief outline of the strategy involved, which relies on the following splitting principle: The fiber F = ℝ^{k} × 0 ⊂ ℝ^{k} × ℝ^{n} splits isometrically as ℝ^{l} × ℝ^{k−l} with the kernel of A^{∗} tangent to the first factor, and the image of A tangent to the second. This kernel extends to the whole ambient space via parallel transport, and corresponds to the translational part of the representation. In other words, the fibration ℝ^{n+k} → M^{n} factors as an orthogonal projection ℝ^{n+k} → ℝ^{n+k−l} followed by a fibration π′ : ℝ^{n+k−l} → M^{n} which is weakly substantial in the sense that the image of the A-tensor spans the whole fiber. π′ thus measures the twisting or rotational part of the representation. The splitting itself is in turn due to a kind of maximum principle: We establish that the polynomial holonomy form has bounded, and therefore parallel derivative.

It will be necessary to first work with parallel horizontal fields along F rather than basic (Bott parallel) ones, and we shall denote the former by lowercase letters, reserving the uppercase notation for basic fields. For a point p in the fiber F, let A_{p} = span{U_{p} | U ∈ A}, where A denotes the space of integrability fields spanned by all A_{X}Y along F. The image of A is then the union of all A_{p} as p ranges over F. Notice also that the kernel of A^{∗} consists of the union of all A_{p}^{⊥}.

By the results in Section 2, the form ω^{♯} is polynomial along every affine plane passing through a point (a, 0) ∈ F spanned by a horizontal x and a vertical u in the image of A. The same is then true for the derivative

∇_{x}ω^{♯} = − ∑_{i} E_{1} ∧ ⋯ ∧ A_{x}^{∗}E_{i} ∧ ⋯ ∧ E_{k}

of ω^{♯} in direction x. If A_{x}^{∗}E_{i} ≠ 0, then the corresponding wedge product in the above expression is nonzero, since A_{x}^{∗}E_{i} is horizontal. But the E_{l} are parallel along F, and A_{x}^{∗}E_{i} is bounded in norm, so that each A_{x}^{∗}E_{i} must be parallel along the geodesic line t ↦ γ_{u}(t) = (a + tu, 0). Thus, for all x, y,

(3.1)  (A_{x}y ∘ γ_{u})′ ≡ 0,  u ∈ im A,

and the image of A, though a priori not of constant rank, is totally geodesic along F, and thus consists of a disjoint union of affine subspaces. The same is true of its orthogonal complement ker A^{∗}: Given u ∈ ker A^{∗}, we claim that γ̇_{u}(t) belongs to the kernel for all t. To see this, consider the variation V (t, s) = exp_{su} tx, which projects down to a variation W = π ∘ V on the quotient. The Jacobi field Y (t) = W_{∗}∂_{s}|_{t,0} induced by W satisfies Y (0) = 0, and

Y′(0) = π_{∗}∇_{∂_{t}}(V_{∗}∂_{s})^{h}|_{(0,0)} = −π_{∗}∇^{h}_{∂_{t}}(V_{∗}∂_{s})^{v}|_{(0,0)} = π_{∗}A_{x}^{∗}u = 0.

Thus, Y is identically 0, or equivalently, the parallel field x is actually basic along γ_{u}, so that A_{x}^{∗}γ̇_{u} = −(x ∘ γ_{u})′ ≡ 0. This establishes the claim.

Up to congruence, A_{0} is 0 × ℝ^{k−l} for some integer l by (3.1). It follows that for any (a, b) ∈ ℝ^{l} × ℝ^{k−l} = F, A_{(a,b)}^{⊥} = ker A_{(a,b)}^{∗} = ℝ^{l} × b, since A_{(0,b)}^{⊥} = ℝ^{l} × b: Indeed, (a, b) ∈ A_{(0,b)}^{⊥}, so that A_{(0,b)}^{⊥} ⊂ A_{(a,b)}^{⊥}, and by symmetry, the reverse inclusion also holds. Thus, A_{(a,b)} = a × ℝ^{k−l}, and F splits isometrically as ℝ^{l} × ℝ^{k−l} with the kernel of A^{∗} tangent to the first factor and the image of A tangent to the second. But the holonomy displacement of ker A^{∗} along horizontal lines γ that intersect F is just parallel translation along γ, so that π factors as an orthogonal projection

ℝ^{l} × ℝ^{n+k−l} → 0 × ℝ^{n+k−l}

followed by a Riemannian submersion π′ : ℝ^{n+k−l} → M^{n}. Furthermore, the latter is weakly substantial in that the totally geodesic fiber F′ over the soul of M is spanned by the image of A. (3.1) then implies that each A_{x}y is a parallel field along F′, or equivalently, that the form Ω is parallel, and therefore also closed. One concludes from (1.1) that for basic X, Y, the integrability field A_{X}Y is a gradient, and having constant norm, A_{X}Y is parallel. As pointed out earlier, the main theorem now follows from [1, Theorem 2.6].

References

[1] D. Gromoll & G. Walschap, Metric fibrations in Euclidean space, Asian J. Math. 1 (1997) 716–728.

State University of New York University of Oklahoma