PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000–000 S 0002-9939(XX)0000-0

THE DUAL FOLIATION IN OPEN MANIFOLDS WITH NONNEGATIVE SECTIONAL CURVATURE

LUIS GUIJARRO AND GERARD WALSCHAP

(Communicated by Jon Wolfson)

Abstract. We study several properties of the Sharafutdinov dual foliation in open manifolds with nonnegative sectional curvature.

1. Introduction

A foliation in a Riemannian manifold M is called metric when its leaves are locally equidistant. A generic Riemannian structure on M does not admit such foliations, except in codimension one (and even then, usually only locally), so it should come as no surprise that the existence of a metric foliation has strong repercussions for the geometry of M.

One of the nicest examples consists of the collection of fibers of a Riemannian submersion π : M → N, for in this case, the leaves are closed, globally equidistant, and the foliation is in fact a fibration, provided M is complete. This occurs in particular for every open manifold M of nonnegative sectional curvature: the metric projection π : M → S of the ambient space onto a soul S which assigns to p ∈ M that point π(p) ∈ S in the soul that is closest to p is a C^{∞} Riemannian submersion (as shown by Perelman for the C^{1} case in [5], one of the authors for the C^{2} case in [2], and by Wilking in [9] for the full regularity). π is also referred to as the Sharafutdinov map, as Perelman proved that it coincides with the retraction constructed in [7].

In [9], Wilking associates to a given metric foliation F another (possibly singular) so-called dual foliation F^{#}, which is also metric whenever M has nonnegative sectional curvature and the leaves are closed. It is therefore natural to look at the foliation dual to the metric projection of an open manifold with nonnegative curvature onto a soul. The main purpose of this note is to study some of its properties.

We will use the standard terminology for metric foliations throughout: F induces an orthogonal splitting T M = V⊕H of the tangent bundle of M, with V (the vertical distribution) denoting the distribution tangent to the leaves, and H (the horizontal distribution) its orthogonal complement. We write e = e^{v}+e^{h} for the corresponding

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

Received by the editors December 14, 2006.

2000 Mathematics Subject Classification. Primary 53C20.

Key words and phrases. Wilking dual foliation, open manifolds with nonnegative sectional curvature.

The first author was supported by research grant MTM2004-04794-MEC. Most of this work was done during a visit of the second author to Madrid, financed in part by funds of the aforementioned grant.

decomposition of a vector e ∈ T M into its vertical and horizontal components. The O’Neill tensor of F is the skew-symmetric tensor field A : H × H → V given by A_{x}y = (1/2)[X, Y]^{v}, with X, Y denoting local horizontal fields extending x, y.

2. The Sharafutdinov dual foliation.

Let M be an open (i.e., complete, noncompact) manifold with nonnegative sectional curvature; recall that Cheeger and Gromoll showed the existence of a compact, totally convex, totally geodesic submanifold S, known as a soul of M, whose normal bundle ν(S) is diffeomorphic to M. As mentioned above, there is a smooth Riemannian submersion π : M → S that is a left inverse to the normal exponential map exp : ν(S) → M, i.e, π ∘ exp(u) = π_{S}(u), where π_{S} : ν(S) → S is the vector bundle projection. Furthermore, recall that if γ̄ : I → S is a geodesic, and U is a parallel section of ν(S) along γ̄, then the map

\begin{matrix}I × ℝ & → M, \\ (t, s) & ↦ R(t, s) := exp(sU(t)) \end{matrix}

is a flat totally geodesic immersion of I ×ℝ into M. Finally, each curve R_{s} : I → M, where R_{s}(t) := R(t, s), is a horizontal geodesic for the submersion π. The reader is invited to consult [3] for further details. We should also point out that since the fiber νp̄(S) of π_{S} over p̄ is a vector space, we often identify tangent vectors to it at different points with elements of νp̄(S), hoping that this will not create any confusion.

2.1. Description of the dual leaf. Recall from [9] that in general, given a metric foliation on a Riemannian manifold M, the dual leaf through a point p consists of all points that can be joined to p by a piecewise smooth horizontal curve. Since any curve can be approximated by piecewise smooth geodesics, it suffices to consider the latter. By the previous paragraph, exponentiating parallel vector fields along curves in the soul yields horizontal curves. On the other hand, uniqueness of horizontal lifts implies that all horizontal curves are obtained in this way. As a result, if p = exp(u), u ∈ ν(S), then the dual leaf F^{#}(p) agrees with the image by exp of the holonomy bundle H(u) through u, which is the subset of the normal bundle of S consisting of all parallel translates of u along curves in S; i.e., F^{#}(p) = exp(H(u)). Of course, as was already noted by Wilking, these leaves stay at a constant distance of the soul.

Our main tool to explore this leaf is the Ambrose-Singer theorem, which in term of vector bundles can be stated as follows: Consider a Euclidean vector bundle E → S over S together with a Riemannian connection, and a point p̄ in S. Parallel translation along piecewise-smooth loops at p̄ in S induces a group of orthogonal transformations of the fiber over p̄, called the holonomy group at p̄. Its Lie algebra 𝔤(p̄) is called the holonomy algebra of the connection at p̄. The theorem says that if R denotes the curvature tensor of the connection, then the holonomy algebra at p̄ is the union over all points q̄ ∈ S of the set of all skew-adjoint endomorphisms Pγ̄R(x̄, ȳ), x̄, ȳ ∈ Tq̄S, where γ̄ is a C^{∞} curve in S joining q̄ to p̄, and Pγ̄ denotes parallel translation along γ̄. But the holonomy algebra through p̄ is the Lie algebra of the holonomy group at p̄, and in our situation, given u ∈ νp̄, the intersection of H(u) with νp̄(S) is precisely the orbit of u under the action of this group. Hence the vertical part of the tangent space to H(u) at u is obtained as 𝔤(p̄) · u, which by the Ambrose-Singer theorem is just the set of all possible vectors of the form Pγ̄R(x̄, ȳ)u. Finally, since the derivative of the normal exponential map sends H̄_{u}, the horizontal space for the normal connection in ν(S), onto the horizontal space H_{p} of π, we conclude that the tangent space of a dual leaf splits as an orthogonal direct sum

(2.1)  T_{p}F^{#}(p) = H_{p} ⊕ exp_{∗u} 𝔤(p̄) · u.

2.2. Identifying local holonomy with radial Jacobi fields. Any v ∈ ν(S) is tangent to a normal geodesic γ_{v} : ℝ → M, which always remains in a π-fiber as we know from Perelman’s theorem ([5]). Denote by q̄ = γ_{v}(0) ∈ S; given any pair of tangent vectors to the soul at q̄, extend them locally in S to obtain fields X̄ and Ȳ with [X̄, Ȳ] = 0, and form the geodesic variations that result from parallel transport of v along the square obtained by following integral curves for X̄, Ȳ, −X̄ and −Ȳ for time \sqrt{s}. It is well known that the Jacobi field J along γ_{v} associated to such variations is the one determined by the initial conditions J(0) = 0, J′(0) = −\frac{1}{2}R(x̄, ȳ)v (see for instance [8]), and therefore collects all the local holonomy information at different points of the soul. Its interaction with the Sharafutdinov submersion π comes from the fact that if A denotes the O’Neill tensor of π, then A_{X}Y ∘ γ_{v} = J, where X and Y are the π-horizontal lifts of x̄ and ȳ along γ_{v}. Thus,

(2.2)  A_{X}Y (exp v) = −\frac{1}{2} exp_{∗v} R(x̄, ȳ)v.

It follows from (2.1) that the tangent space to a dual leaf not only contains the space generated by the horizontal subspace and the image of the A-tensor (this was already implicit in [9]), but is in general strictly larger than it. We describe it in more detail below.

2.3. Holonomy diffeomorphisms and the dual leaf. For convenience, in what follows, we denote a π-fiber π^{−1}(q̄) by Fq̄. Any curve ᾱ : [a, b] → S induces a diffeomorphism hᾱ between the π-fibers at its endpoints: given a point q ∈ F_{ᾱ(a)}, hᾱ(q) is defined to be the endpoint of the horizontal lift of ᾱ that starts at q. hᾱ is called the holonomy diffeomorphism induced by ᾱ. Since horizontal curves are obtained by exponentiating parallel sections in the normal bundle, we have that

(2.3)  hᾱ ∘ exp = exp ∘Pᾱ.

From the construction of F^{#}, it is clear that hᾱ maps F^{#} ∩ F_{ᾱ(a)} to F^{#} ∩ F_{ᾱ(b)}, and therefore its derivative preserves the vertical tangent space to the leaves of F^{#}. This enables us to give a version of the Ambrose-Singer theorem for leaves:

Lemma 2.1. Let p = exp(u) for some u ∈ ν(S). For any curve ᾱ : [a, b] → S with ᾱ(b) = π_{S}(u), denote by hᾱ its holonomy diffeomorphism, and by α its horizontal lift with α(b) = p. Then the set

VF_{p}^{#} := { h_{ᾱ∗}(A_{X}Y (α(a))) : for all such ᾱ’s, all x̄, ȳ ∈ T_{ᾱ(a)}S }

agrees with the π-vertical tangent subspace to F^{#}(p) at p.

Proof. By (2.1) the vertical component of the tangent space of F^{#}(p) at p is exp_{∗u} 𝔤(p) · u, which consists of all vectors of the form exp(PᾱR(x̄, ȳ))u, with ᾱ, x̄, and ȳ as in the statement. Under the identification of a vector space with its tangent space, P_{ᾱ∗} = Pᾱ since the latter is linear, and by (2.2) and (2.3),

exp_{∗u}(PᾱR(x̄, ȳ))u = exp_{∗u} Pᾱ (R(x̄, ȳ)(P −1 ; ᾱu)) = exp_{∗u} ∘P_{ᾱ∗} (R(x̄, ȳ)(P −1 ; ᾱu))

= h_{ᾱ∗} ∘ exp_{∗} R(x̄, ȳ)(P −1 ; ᾱu) = −2h_{ᾱ∗}A_{X}Y (α(a)).

□

It is also clear that the derivative of a holonomy diffeomorphism h preserves the normal bundle of leaves in F^{#}, on which it is in fact isometric. The reason for this is given in [9], where it is shown that the normal space of F^{#} along a horizontal geodesic is spanned by parallel Jacobi fields.

2.4. The pullback bundle. For u ∈ ν(S), consider the holonomy bundle H(u) through u. The restriction of the projection π_{S} to H(u) (which will be denoted by Π) allows us to construct the pullback bundle Π^{∗}ν(S). Observe that this bundle always splits off a line bundle ℝ, namely the one generated by the base point in H(u) (more precisely, the total space of the pullback bundle is the set of pairs (v, w), where v ∈ H(u) and π_{S}(v) = π_{S}(w); the line bundle has as total space {(v, sv) | v ∈ H(u), s ∈ ℝ} . It follows that whenever the dual leaf through p = exp(u) does not fill the whole boundary of a tubular neighbourhood of the soul, then Π^{∗}ν(S) is reducible even further:

Lemma 2.2. In the above situation, Π^{∗}ν(S) = ℝ ⊕ 𝔤 · ℝ ⊕ N where • ℝ is the line bundle generated by v at each point v ∈ H(u);

• 𝔤 · ℝ is the bundle whose fiber at each v ∈ H(u) is 𝔤(π_{S}(v)) · ℝv;

• N is the orthogonal complement of the first two summands.

The bundle N could of course also be replaced by any other transversal complement to the first two summands.

3. Applications

In this section we present some consequences of the above discussion:

3.1. Quasipositive curvature. Let B_{r}(S) be a tubular neighborhood of radius r about the soul. For small r > 0 this set is convex, and therefore its boundary S_{r} inherits a metric of nonnegative curvature. Since any horizontal geodesic remains at constant distance from S, S_{r} is composed entirely of dual leaves to the Sharafutdinov foliation.

Theorem 3.1. If the induced metric on some S_{r} has positive sectional curvature at one point, then the normal holonomy of the soul acts transitively on the sphere bundle; namely, for any two vectors u, v ∈ ν(S) with ∥u∥ = ∥v∥, there is a curve c : I → S in the soul such that P_{c}(u) = v. In particular, the normal exponential map exp : ν(S) → M is a diffeomorphism.

Proof. S_{r} can contain one or more fibers of the dual foliation. In the first case, the holonomy acts transitively by the discussion in the previous section, and we are done. So assume S_{r} contains more than one leaf. Let F^{#}(p) be the leaf passing through the positive curvature point p. The normal subspace to F^{#}(p) at p splits into the direct sum of a radial direction orthogonal to S_{r} and a nonempty subspace V_{p} tangent to S_{r} that generates parallel Jacobi vector fields along horizontal geodesics by Theorem 6.1 in [9]. Thus if x ∈ H_{p} and v ∈ V_{p}, then the extrinsic curvature of the plane spanned by x and v is zero. It is clear from the Gauss equation that the same is true for the sectional curvature of the induced metric in S_{r}, since the normal vector to S_{r} is parallel along the horizontal geodesic tangent to x. This of course contradicts our assumption on the curvature of S_{r}. The reason why the normal exponential map is a diffeomorphism is because the parallel translate of a ray direction (i.e., of a unit vector u such that the geodesic t ↦ exp(tu) is minimal on [0, s] for all s > 0) is again a ray direction. Since there is always at least one ray direction, every direction must then be such. □

Another standard consequence of having transitive holonomy is that the ideal boundary of M is a point (see for instance [4]). The novelty in our situation is that the assumption involves the sectional curvature at only one point, suggesting a previously unknown strong rigidity in nonnegative curvature.

3.2. The normal cut and conjugate locii for the soul and the dual foliation. Dual leaves are constructed by means of horizontal geodesics, and the latter are easily described by parallel translation and the normal exponential map. One would therefore expect the singularities of the exponential map to be reflected in the dual leaves:

Theorem 3.2. If p ∈ F^{#}(p) is a conjugate or cut point for S, then the same is true of any other point in F^{#}(p).

Proof. Given any q ∈ F^{#}(p), there exists some piecewise geodesic ᾱ : I → S whose horizontal lift α to p ends at q. Let u ∈ ν(S) with exp(u) = p such that the parallel transport of u along ᾱ (which we denote by U) exponentiates to α. If p is a conjugate point of exp : ν(S) → M, then there is some v ∈ ν(S) with exp_{u∗}(v) = 0. By (2.3), exp_{P_{ᾱ(u)}∗}(Pᾱv) = h_{∗} ∘ exp_{u∗}(v) = 0, so that q is a conjugate point of S along the geodesic tangent to P_{α}(u).

To show the second part of the theorem, recall that p is a a cut point of the soul if it is a cut point of π(p) along some geodesic γ. This means that if p = γ(t_{0}), then for any ε > 0 there exists a shorter geodesic c^{ε} connecting π(p) to γ(t_{0} + ε). Obviously these geodesics biangles are preserved under the holonomy diffeomorpshims h, and therefore q is a cut point of π(q) along h ∘ γ. □

3.3. The Cheeger-Gromoll convex function. The main tool in the proof of the Soul Theorem is the construction of a convex function with compact sublevel sets [1]. This can also be applied to any nonnegatively curved Alexandrov space. If the latter is taken to be the “quotient” space obtained from the leaf closures of the dual foliation, then there is a direct relation between the two functions. These facts are stated in a more precise manner in section 7 of [9], but since we will need them later, we include a version that suffices to our needs here:

Theorem 3.3 (Wilking, 06). Let M be an open manifold with nonnegative sectional curvature. If S is a soul of M, then there exists a noncompact Alexandrov space A and a submetry σ : M → S × A, where the target is endowed with the product metric. If π_{A} : S ×A → A denotes projection onto the second factor, then the fibers of π_{A} ∘ σ agree with the closures of the dual leaves.

We can now explicitly describe the relation between the convex functions:

Proposition 3.4. Let b : M → ℝ be the Cheeger-Gromoll function constructed at the initial point p, and b_{A} : A → ℝ that constructed at σ(p). Then b = b_{A} ∘ π_{A} ∘ σ.

Denote by σ_{A} = π_{A} ∘ σ. We prove the above claim together with:

Proposition 3.5. If γ : [0, ∞) → M is a ray, then σ_{A} ∘ γ is a ray in A.

Proof. By Remark 8.3.b in [9], σ_{A} takes level sets of b to level sets of b_{A}. Since b(p) = b_{A}(σ_{A}(p)) = 0, Proposition 3.4 follows. On the other hand, it is well known that γ is a ray if and only if b(γ(t)) = b(γ(0)) + t for all t > 0 and similarly for A. Since b_{A}(σ_{A}(γ(t))) = b(γ(t)) = t + b(γ(0)) = t + b_{A}(σ_{A}(γ(0))), we have that σ_{A} ∘ γ is a ray in A. □

The submetry σ_{A} and the space A are interesting in their own right, since they contain much of the large scale information of M. We point out a few of their properties:

• If p = exp(u), Wilking’s definition implies that σ_{A}^{−1}(σ_{A}(p)) = \overline{F^{#}(p)}; its tangent space splits as H_{p} ⊕exp_{∗u} \overline{𝔤(p̄) · u}; therefore, the horizontal tangent space for σ_{A} coincides with the π-vertical directions inducing parallel Jacobi fields along π-horizontal geodesics. Thus any geodesic in A lifts to geodesics entirely contained in Sharafutdinov fibers by [9].

• As Wilking points out in [9], remark 8.3(a), the space A is bilipschitz equivalent to the quotient ν_{p}(S)/\overline{H} of the normal space ν_{p}(S) to S at p ∈ S by the closure of the holonomy group H of the normal bundle of S. In fact, if we denote by F_{p}/\overline{H} the image of this set by the exponential map, then A is isometric to any quotient F_{π(p)}/\overline{H}, for p ∈ M. This is hardly surprising, since the holonomy diffeomorphisms in M are isometries on the σ_{A}-horizontal subspaces.

• The ideal boundaries of A and M are isometric. To see this, notice that for any positive λ, σ_{A} induces a submetry σ_{A}^{λ} : λM → λA; now choose a point p ∈ S and σ_{A}(p) ∈ A; usual Gromov-Hausdorff arguments for pointed metric spaces show that there is a limiting submetry σ_{A}^{∞} : C_{0}(M(∞)) → C_{0}(A(∞)), where C_{0} denotes the Euclidean cone. If γ : [0, ∞) → M is a ray starting at p, it is easy to see that the fibers of σ_{A} along the points γ(t) all share a common upper bound for their diameter, since they are obtained by exponentiating the closure of the holonomy orbit of the vectors γ̇(t). Therefore, in the limit the submetry σ_{A}^{∞} becomes bijective, hence an isometry.

3.4. The Sharafutdinov retraction preserves dual leaves. As mentioned earlier, Sharafutdinov constructed in [7] a retraction of M onto S mapping each p ∈ M to π(p). This map was used later by Perelman in his proof of the soul conjecture of Cheeger and Gromoll. The retraction is given by a map H : [0, 1] × M → M, and each curve H(·, p) : [0, 1] → M is called a Sharafutdinov line. It is often more convenient to use outgoing Sharafutdinov lines (OSL) instead (see [3] for a definition and properties needed for this section). Specifically, a curve t ↦ H(t, p), t ∈ [0, 1], is nothing more than an OSL c : [0, a] → M with c(0) = H(1, p), c(a) = p.

It turns out that dual leaves are preserved at each stage of this retraction:

Theorem 3.6. Suppose p, q ∈ F^{#}(p). If t ∈ [0, 1], then H(t, p) and H(t, q) belong to the same dual leaf.

Proof. In [3], it is shown that if h is a holonomy diffeomorphism and c : [0, ∞) → M is an OSL starting at p, then h ∘ c is an OSL at h(p). Therefore, H(t, h(p)) = h∘H(t, p) by the property mentioned in the above paragraph, and the claim follows from the fact that any two points in a dual leaf are joined by a piecewise smooth horizontal geodesic. □

Perelman constructed a Sharafutdinov-like retraction for open Alexandrov spaces with nonnegative curvature [6]. It turns out that if A is the Alexandrov space from Theorem 3.3, then the submetry σ_{A} : M → A preserves that construction:

Theorem 3.7. Let c : [0, 1] → M be the Sharafutdinov line with c(0) = p, c(1) ∈ S. Then c̄ := σ_{A} ∘ c : [0, 1] → A is the unique Sharafutdinov line with c̄(0) = σ_{A}(p), c̄(1) ∈ σ_{A}(S).

Proof. According to [6], the Sharafutdinov lines are left tangent to a “gradient” vector field of b: namely, for p ∈ M with b(p) > 0, the set C = b^{−1}(b(p)) is totally convex and thus an Alexandrov space with nonnegative curvature. Its space of directions at p is an Alexandrov space with curvature greater than or equal to 1, and hence contains a unique “soul”; i.e, a direction at maximal distance from its boundary in the tangent space. Renormalizing this field suitably, it can be shown that it has left continuous integral curves tracing the Sharafutdinov retraction.

In our case, and from the arguments in [6], it remains to show that the Sharafutdinov fields of M and A are related by means of the derivative of σ_{A}. However, this is clear from the description of the tangent spaces of the C_{t}’s given in [10], together with the fact that the σ_{A}-images of the C_{t}’s form precisely the convex exhaustion of A. □

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Department of Mathematics, Universidad Autónoma de Madrid.

E-mail address: luis.guijarro@uam.es

Department of Mathematics, University of Oklahoma. E-mail address: gerard@ou.edu