THE SMOOTHNESS OF RIEMANNIAN SUBMERSIONS

WITH NONNEGATIVE SECTIONAL CURVATURE

Jianguo Cao and Mei-Chi Shaw

In this article, we study the smoothness of Riemannian submersions for open manifolds with non-negative sectional curvature. Suppose that M^{n} is a C^{∞}-smooth, complete and non-compact Riemannian manifold with nonnegative sectional curvature. Cheeger-Gromoll [ChG] established a fundamental theory for such a manifold. Among other things, they showed that M^{n} admits a totally convex exhaustion {Ω_{u}}_{u≥0} of M^{n}, where Ω_{0} = S is a totally geodesic and compact submanifold without boundary. Furthermore, M^{n} is diffeomorphic to the normal vector bundle of the soul S.

Sharafutdinov found that there exists a distance non-increasing retraction Ψ : M^{n} → S from the open manifold M^{n} of non-negative sectional curvature to its soul, (cf. [Sh], [Y2]). Perelman [Per] further showed that such a map Ψ is indeed a C^{1}-smooth Riemannian submersion. Furthermore, Ψ[Exp_{q}(tv⃗)] = q for any q ∈ S and v⃗⊥T_{q}(S). Therefore, the fiber F_{q} = Ψ^{−1}(q) is a k-dimensional submanifold, which is C^{∞}-smooth almost everywhere, where k = dim(M^{n}) − dim(S) > 0.

Guijarro [Gu] proved that the fiber F_{q} is indeed a C^{2}-smooth submanifold for each q ∈ S. In this paper, we prove that the fibres are C^{∞}-smooth.

Theorem 1. Let M^{n} be a complete, non-compact and C^{∞}-smooth Riemannian manifold with nonnegative sectional curvature. Suppose S is a soul of M^{n}. Then any distance non-increasing retraction Ψ : M^{n} → S must give rise to a C^{∞}-smooth Riemannian submersion with the property that

Ψ[Exp_{q}(ℝ{v⃗})] = q

for all q ∈ S and v⃗⊥S, where Exp_{q} is the exponential map of M^{n} at q.

Professor Wilking kindly informed us that he has recently obtained the similar result (cf. [Wi]). His method is completely independent of ours. Our proof of

Typeset by AMS-TEX

Both authors are supported by NSF Grants. Research at MSRI is supported in part by NSF grant DMS-9810361. The first author is grateful to MSRI and Max-Planck Institute for Mathematics at Leipzig for their hospitality.

Theorem 1 uses a flat strip theorem associated with Cheeger-Gromoll exhaustion (cf. [CaS] or Theorem 4 below), an uniform estimate for cut-radii of convex subsets in [ChG] and a bootstrap argument using the smoothness of geodesic flow.

For each compact convex subset Ω ⊂ M^{n}, we let U_{ϵ}(Ω) = {x ∈ M^{n}|d(x, Ω) < ϵ}. Its cut-radius is given by δ_{Ω} = sup{ϵ| there is a unique nearest point projection P_{Ω} : U_{ϵ}(Ω) → Ω}.

For each x ∈ M^{n}, we let Inj_{M^{n}}(x) be the injectivity radius of M^{n} at x. Similarly, let Inj_{M^{n}}(A) = {Inj_{M^{n}}(x)|x ∈ A}. We need the following semi-global estimate for cut-radius. Recall that there is a totally convex exhaustion {Ω_{u}}_{u≥0} of M^{n} given in [ChG].

Lemma 2. (Lemma 2.4 of [ChG], [CaS]) Let A ⊂ Ω_{T} be a connected, convex and compact subset in a Riemannian manifold M^{n} with nonnegative curvature, let K_{0} = max{K(x)|x ∈ Ω_{T +1}}, Inj_{M^{n}}(Ω_{T}) be the upper bound of sectional curvature on Ω_{T +1} and S be as above. Suppose that dim(Ω_{T}) = n. Then the subset A has cut-radius bounded below by

δ_{A} ≥ δ_{0}(T) = \frac{1}{4} min\{Inj_{M^{n}}(Ω_{T}), \frac{π}{\sqrt{K_{0}}} , 1\},

where δ_{0}(T) is independent of choices of A with A ⊂ Ω_{T}.

Let us briefly recall the Cheeger-Gromoll convex exhaustion. According to [ChG], there is a partition a_{0} = 0 < a_{1} < ...a_{m} < a_{m+1} = ∞ of [0, ∞) and an exhaustion of {Ω_{u}}_{u≥0} of M^{n} such that the following holds:

(1) M^{n} = ∪_{u≥0}Ω_{u}. If u > a_{m} then dim[Ω_{u}] = n. If u ≤ a_{m}, then dim[Ω_{u}] < n.

(2) Ω_{0} = S is the soul of M^{n}, which is a totally geodesic C^{∞}-smooth compact submanifold without boundary.

(3) If u > 0, Ω_{u} is a totally convex, compact subset of M^{n} and hence Ω_{u} is a compact submanifold with a C^{∞}-smooth relative interior. Furthermore, dim(Ω_{u}) = k_{u} > 0 and Ω_{u} has a non-empty (k_{u} − 1)-dimensional relative boundary ∂Ω_{u};

(4) For any u_{0} ∈ [a_{j}, a_{j+1}] and 0 ≤ t ≤ u_{0} − a_{j}, the family {Ω_{u_{0}−t}}_{t∈[0,u_{0}−a_{j}]} is given by the inward equidistant evolution:

Ω_{u_{0}−t} = {x ∈ Ω_{u_{0}}|d(x, ∂Ω_{u_{0}}) ≥ t}.  (2.1)

(5) If u > a_{m} then u − a_{m} = max{d(x, ∂Ω_{u})|x ∈ Ω_{u})}. If 0 ≤ j ≤ m − 1 then a_{j+1} − a_{j} = max{d(x, ∂Ω_{a_{j+1}})|x ∈ ∂Ω_{a_{j+1}}} and hence dim[Ω_{a_{j}}] < dim[Ω_{a_{j+1}}] for j ≥ 0.

Assume that k = dim[M^{n}] − dim[S] = dim(F_{q}) for all q ∈ S. Since M^{n} = ∪_{T ≥0}Ω_{T}, it is sufficient to verify that the subset [U_{δ_{0}(T)}(Ω_{T})∩F_{q}] has a k-dimensional C^{∞}-smooth interior, where δ_{0}(T) is given by Lemma 2 and T > a_{m}.

For this purpose, we need to study the geometry of the equidistant hypersurfaces from ∂Ω_{u}. Federer [Fe] has studied the smoothness of the outward equidistant hypersurfaces ∂[U_{ϵ}(Ω)] for 0 < ϵ < δ_{Ω}. Following his approach, we consider the outward normal cone of Ω as follows:

N^{+}(Ω, M^{m}) = {(p, v⃗)|p ∈ Ω, d(Exp_{p}(tv⃗), Ω) = t|v⃗|, for 0 ≤ t|v⃗| < δ_{Ω}}.

If {Ω_{u}} is the Cheeger-Gromoll inward exhaustion as above and u > 0, then the relative boundary ∂Ω_{u} is not necessarily smooth. We are going to study the corresponding decomposition of N^{+}(Ω, M^{m}):

N_{p}^{+}(Ω_{u}, M^{n}) ⊂ [N_{p}^{+}(Ω_{u}, int(Ω_{u+ϵ})) ⊕ N_{p}^{+}(int(Ω_{u+ϵ}), M^{n})],  (2.2)

where N_{p}^{+}(Ω_{u}, int(Ω_{u+ϵ})) is defined by

N^{+}(Ω_{u}, int(Ω_{u+ϵ})) ={(p, v⃗)|p ∈ Ω_{u}, d(Exp_{p}(tv⃗), Ω_{u}) = t|v⃗|,
for 0 ≤ t|v⃗| < δ_{Ω_{u}}, Exp_{p}(tv⃗) ∈ int(Ω_{u+ϵ})}.

Our next step is to choose ϵ sufficiently small so that (1) there is a nearest point projection P : int(Ω_{u+ϵ}) → Ω_{u}; and (2) Ω_{u} = {x ∈ Ω_{u+ϵ}|d(x, ∂Ω_{u+ϵ}) ≥ ϵ} holds. We first find j so that a_{j} ≤ u < a_{j+1} for some 0 ≤ j ≤ m. Let T = u + a_{m} + 1 and δ_{0}(T) be given by Lemma 2. It follows from a result of Yim that there is a constant C_{T} such that, for 0 ≤ u_{1} < u_{2} ≤ T, we have

max{d(x, Ω_{u_{1}})|x ∈ Ω_{u_{2}}} ≤ C_{T} (u_{2} − u_{1}),  (2.3)

see [Y2, Theorem A.5(3)].

In what follows, we always choose

0 < ϵ = ϵ_{u} < min\{[a_{j+1} − u], \frac{δ_{0}(T)}{2C_{T}}\},  (2.4)

where u ∈ [a_{j}, a_{j+1}), T = u + a_{m} + 1 and δ_{0}(T) is given by Lemma 2.

With such a choice of ϵ = ϵ_{u} by (2.4), the geometry of N_{p}^{+}(Ω_{u}, int(Ω_{u+ϵ})) is determined by its minimal normal vectors which we now describe.

Definition 3. (Minimal normal vector) Let Ω_{u}, Ω_{u+ϵ} and N^{+}(Ω_{u}, int(Ω_{u+ϵ})) be as above. Let σ_{(p,v⃗)} : [0, ϵ] → M^{n} be a geodesic given by σ_{(p,v⃗)}(t) = Exp_{p}(t\frac{v⃗}{|v⃗|}), where v⃗ ≠ 0. If σ_{(p,v⃗)} is a length-minimizing geodesic from p ∈ Ω_{u} to ∂Ω_{u+ϵ}, then v⃗ is called a minimal normal vector in N_{p}^{+}(Ω_{u}, int(Ω_{u+ϵ})).

It is known that any other normal vector w⃗ ∈ N_{p}^{+}(Ω_{u}, int(Ω_{u+ϵ})) can be expressed as a linear combination of minimal normal vectors at p. Moreover, the convex hull of minimal normal vectors at p is equal to N_{p}^{+}(Ω_{u}, int(Ω_{u+ϵ})), (cf. [Y1, Proposition 1.7]).

For each p ∈ M^{n}, we let V_{p} = T_{p}(F_{Ψ(p)}) and H_{p} = [V_{p}]^{⊥}. A geodesic α : [a, b] → M^{n} is said to be horizontal, if α′(t)⊥F_{Ψ(α(t))} for all t ∈ [a, b]. We need the following flat strip theorem for the proof of Theorem 1.

Theorem 4. Let {Ω_{u}} be the Cheeger-Gromoll totally convex exhaustion of M^{n} as above. Suppose that Ψ : M^{n} → S be a distance non-increasing retraction and F_{q} = Ψ^{−1}(q) be a fibre for some q ∈ S. Then for p ∈ F_{q} ∩ Ω_{u} and any (p, v⃗) ∈ N^{+}(Ω_{u}, M^{n}), we have

Ψ[Exp_{p}([ℝ{v⃗}])] = Ψ(p) = q.  (4.1)

Moreover, if dim(S) ≥ 1 and if w⃗ ∈ H_{p} has |w⃗| = 1 = |v⃗|, then the surface Σ_{v⃗,w⃗}^{2} = Exp_{p}[ℝ{v⃗} ⊕ ℝ{w⃗}] is totally geodesic immersed flat plane in M^{n}.

A result similar to Theorem 4 was proved in [CaS] via a totally different approach, which involves the barrier surface technique inspired by the work of Calabi.

Proof of Theorem 4. Theorem 4 was proved by Perelman [Per] for the case of Ω_{0} = S. Applying Perelman’s argument for the case of p ∉ S, Guijarro [Gu1] found the following sufficient condition for (4.1).

(4.2) v⃗ ∈ V_{p} stays vertical under parallel transport along any horizontal broken geodesic.

Guijarro showed that (4.1) follows from (4.2). Moreover, if (4.2) holds and if w⃗ ∈ H_{p} has |w⃗| = 1 = |v⃗|, then the surface Σ_{v⃗,w⃗}^{2} = Exp_{p}[ℝ{v⃗} ⊕ ℝ{w⃗}] is totally geodesic immersed flat plane in M^{n}, (cf. Theorem 3.1 of [Gu1]).

In order to see that N^{+}(Ω_{u}, M^{n}) ⊂ V_{p} holds, we also need

(4.3) ([Gu2, Corollary 3.2])Any horizontal geodesic α with α(0) ∈ ∂Ω_{u} must be entirely contained in ∂Ω_{u}. Consequently, H_{p} ⊂ T_{p}^{−}(∂Ω_{u}), where T_{p}^{−}(∂Ω_{u}) is the tangent cone of ∂Ω_{u} at p

For the convenience of the reader, we include Guijarro’s short proof here. Let ℓ = d(α(0), S). It follows from Perelman’s Theorem that α(ℝ) ⊂ Ū_{ℓ}(S). Choose sufficient large N > a_{m} so that Ω_{N} ⊃ Ū_{ℓ+1}(S). By Theorem 1.10 of [ChG], the distance function r_{N}(y) = d(y, ∂Ω_{N}) is convex for y ∈ Ω_{N}. The function h_{N}(s) = r_{N}(α(s)) is a bounded convex function for s ∈ ℝ and hence h must be a constant function. For u > a_{m}, then h_{N}(s) ≡ c = [N − u] and hence α(ℝ) ⊂ ∂Ω_{N−(N−u)} = ∂Ω_{u}. For u ≤ a_{m}, we have h_{N}(s) ≡ c = [N − a_{m}] and α(ℝ) ⊂ Ω_{a_{m}}. Let j be the integer such that a_{j+1} ≥ u > a_{j}. Consider the distance functions r_{a_{i}}(y) = d(y, ∂Ω_{a_{i}}) for i = m, m − 1, ..., j + 1. For the same reason as above, one can show that the assertion (4.3) holds for any u.

Let T = u + a_{m} + 1 and δ_{0}(T) and ϵ be in (2.4) as above. Recall that by (2.2) we have

N_{p}^{+}(Ω_{u}, M^{n}) ⊂ [N_{p}^{+}(Ω_{u}, int(Ω_{u+ϵ})) ⊕ N_{p}^{+}(int(Ω_{u+ϵ}), M^{n})].

For v⃗ in either N_{p}^{+}(Ω_{u}, int(Ω_{u+ϵ})) or N_{p}^{+}(int(Ω_{u+ϵ}), M^{n}), we will show that such a v⃗ satisfies (4.2).

It follows from Theorem 1.10 of [ChG] (or Corollary 1.4 of [Y1]) that any minimal normal vector v⃗ of N_{p}^{+}(Ω_{u}, int(Ω_{u+ϵ})) stays minimal under parallel transport along any geodesic in ∂Ω_{u}. Hence, the bundle N^{+}(Ω_{u}, int(Ω_{u+ϵ})) is invariant under parallel transport along any geodesic in ∂Ω_{u}. This together with (4.3) implies that if v⃗ ∈ N_{p}^{+}(Ω_{u}, int(Ω_{u+ϵ})) then v⃗ satisfies (4.2).

For v⃗ ∈ N_{p}^{+}(int(Ω_{u+ϵ}), M^{n}), the assertion (4.2) follows from Corollary 3.2 of [Gu1]. In fact, since int(Ω_{u+ϵ}) is totally convex and totally geodesic, both T (int(Ω_{u+ϵ})) and N^{+}(int(Ω_{u+ϵ}), M^{n}) are invariant under parallel transport along any geodesic in int(Ω_{u+ϵ}). This together with (4.3) implies that (4.2) holds for any v⃗ ∈ N_{p}^{+}(int(Ω_{u+ϵ}), M^{n}).

Therefore, (4.2) holds for any v⃗ ∈ N_{p}^{+}(Ω_{u}, M^{n}). This completes the proof of Theorem 4. □

For the proof of Theorem 1, we consider the geodesic flow of the unit tangent sphere bundle

SM^{n} = {(p, v⃗)|p ∈ M^{n}, v⃗ ∈ T_{p}(M^{n}), |v⃗| = 1}.

The geodesic flow τ_{t} : SM^{n} → SM^{n} is given by τ_{t}(p, v⃗) = (Exp_{p}(tv⃗), \frac{d[Exp_{p}(tv⃗)]}{dt}). When the metric of M^{n} is C^{∞}-smooth, τ_{t} is a C^{∞}-smooth diffeomorphism.

With Lemma 2, Theorem 4 and the smoothness of the geodesic flow, we are ready to prove Theorem 1.

Proof of Theorem 1. As we pointed above, it is sufficient to verify that the subset [U_{δ_{0}(T)}(Ω_{T}) ∩ F_{q}] has a k-dimensional C^{∞}-smooth interior for any given T > a_{m} and q ∈ S, where δ_{0}(T) is given by Lemma 2.

Fix T > a_{m} with dim[Ω_{T}] = n. For each u ∈ [0, T], there is a distance function r_{u}(y) = d(y, Ω_{u}). Let int(Ω_{u}) = [Ω̄_{u} − ∂Ω_{u}] be the relative interior of Ω_{u}. Recall that int(Ω_{u}) is a totally convex and totally geodesic submanifold of M^{n}. Hence, int(Ω_{u}) is a C^{∞}-smooth submanifold of M^{n}. Although the level set r_{u}^{−1}({δ}) may not be C^{∞}-smooth, the sub-level set r_{u}^{−1}((−∞, δ)) is always an open C^{∞}-smooth domain of M^{n}. We are going to verify the following assertion:

Assertion (5.u). The sub-level set [U_{δ_{0}(T)}(Ω_{u}) ∩ F_{q}] has the k-dimensional C^{∞}- smooth interior, where q ∈ S and k = dim[F_{q}].

Since the soul S has the cut radius ≥ δ_{0}(T) and S is C^{∞}-smooth, the assertion (5.u) holds for u = 0 and Ω_{0} = S by Perelman’s theorem.

Our next step is to show that the assertion (5.u) holds for 0 ≤ u ≤ \frac{δ_{0}(T)}{2C_{T}}, where C_{T} is given by (2.3).

Since Ω_{u} is a compact subset and U_{δ_{0}(T)}(S) is an open subset in M^{n}, we see that ℓ_{u} = max\{d(x, S)|x ∈ Ω_{u}\} ≤ \frac{δ_{0}(T)}{2}. Hence, we have ϵ_{0} = δ_{0}(T) − ℓ_{u} > 0.

For each x ∈ [U_{δ_{0}(T)}(Ω_{u})∩F_{q}−U_{δ_{0}(T)}(S)], we are going to choose a k-dimensional open neighborhood V_{ϵ}(x) of x in F_{q} and a C^{∞}-smooth coordinate system for V_{ϵ}(x).

The construction of V_{ϵ}(x) takes two steps. Let P_{u} : U_{δ_{0}(T)}(Ω_{u}) → Ω_{u} be the nearest point projection from M^{n} to Ω_{u} and y_{x} = P_{u}(x). Suppose that σ : [0, ℓ_{x}] → M^{n} is the unique shortest geodesic segment of unit speed from y_{x} to x with σ(0) = y_{x} and σ(ℓ_{x}) = x. Since σ′(0) ∈ N^{+}(Ω_{u}, M^{n}), it follows from Theorem 4 that σ is entirely contained in F_{q}.

Let us now begin to construct V_{ϵ}(x). Recall that ϵ_{0} = δ_{0}(T) − ℓ_{u} > 0. We choose 0 < ϵ_{1} < ϵ_{2} < ϵ_{0} and an annular region:

A(Ω_{u}, ϵ_{1}, ϵ_{2}) = {z ∈ M^{n}|ϵ_{1} < d(z, Ω_{u}) < ϵ_{2}}

It is clear that A(Ω_{u}, ϵ_{1}, ϵ_{2}) ⊂ U_{δ_{0}(T)}(S). We already showed that F_{q}∩A(Ω_{u}, ϵ_{1}, ϵ_{2}) ⊂ F_{q} ∩ U_{δ_{0}(T)}(S) is C^{∞}-smooth k-dimensional open sub-manifold.

Let us choose ϵ_{3} with ϵ_{1} < ϵ_{3} < ϵ_{2} and x̂ = σ(ϵ_{3}). We consider r(y) = d(y, Ω_{u}). By Corollary 5.3 of [Fed], the level set [r^{−1}(ϵ_{3}) ∩ F_{q}] is a C^{1,1}-smooth (k − 1)- dimensional hypersurface in [F_{q} ∩ A(Ω_{u}, ϵ_{1}, ϵ_{2})]. The integral curves of the gradient of ∇r are portions of normal geodesics from Ω_{u}. It follows from Theorem 4 that those integral curves (with initial points in [F_{q} ∩ A(Ω_{u}, ϵ_{1}, ϵ_{2})]) must be entirely contained in F_{q}. Hence, ∇r is tangent to F_{q} on the smooth subset [F_{q}∩A(Ω_{u}, ϵ_{1}, ϵ_{2})]. Let Σ^{k} = [F_{q} ∩A(Ω_{u}, ϵ_{1}, ϵ_{2})] and B^{k−1}(0) be the unit ball in ℝ^{k−1}. It is well-known (cf. [Hi]) that the mapping space C^{∞}(B^{k−1}(0), Σ^{k}) is dense in another mapping space C^{1,1}(B^{k−1}(0), Σ^{k}). Therefore, there exist an element h ∈ C^{∞}(B^{k−1}(0), Σ^{k}) and a C^{∞}-smooth hypersurface W^{k−1} = h(B^{k−1}(0)) such that (1) the smooth hypersurface W^{k−1} is C^{1}-close to the C^{1,1}-hypersurface [r^{−1}(ϵ_{3}) ∩ F_{q} ∩ U_{ϵ_{4}}(x̂)] in a sufficiently small neighborhood F_{q} ∩ U_{ϵ_{4}}(x̂) of x̂; and (2) W^{k−1} is transversal to the vector field ∇r everywhere. Thus, there is a C^{∞}-smooth local coordinate system around x̂:

ĥ : B_{ϵ}^{k−1}(0) × (t_{1}, t_{2}) → V̂_{ϵ}(x̂)

such that \frac{∂ĥ}{∂t} = ∇r and the curve t → ĥ((w_{1}, ..., w_{k−1}), t) is a integral curve of ∇r. The vector \frac{∂ĥ}{∂w_{j}} is transversal to \frac{∂ĥ}{∂t}, but \frac{∂ĥ}{∂w_{j}} is not necessarily orthogonal to \frac{∂ĥ}{∂t} for j = 1, 2, ..., k − 1.

Notice that V̂′ = \{(ĥ(y), \frac{∂ĥ}{∂t}(y))|y ∈ B_{ϵ}^{k−1}(0) × (t_{1}, t_{2})\} is a k-dimensional, C^{∞}-smooth submanifold of SM^{n}. If c = d(x, x̂), then τ_{c}(V̂′) remains to be a kdimensional C^{∞}-smooth submanifold of SM^{n}, where {τ_{t}}_{t∈ℝ} is the geodesic flow of the unit tangent sphere bundle SM^{n}. We may assume that ϵ_{3} +c+|t_{2} −t_{1}| < δ_{0}(T) by choosing |t_{2} − t_{1}| sufficiently small.

Let π : SM^{n} → M^{n} be the foot-point projection given by π(p, v⃗) = p. By Theorem 4, we have τ_{c}(V̂′) ⊂ SF_{q}, where SF_{q} is the unit tangent sphere bundle of the fibre F_{q}. Since ϵ_{3} + c + |t_{2} − t_{1}| < δ_{0}(T) and Ω_{u} has cut-radius ≥ δ_{0}(T), the restriction π|_{τ_{c}(V̂′)} : τ_{c}(V̂′) → F_{q} is injective.

Setting V_{ϵ}(x) = π[τ_{c}(V̂′)], we are going to show that V_{ϵ}(x) is diffeomorphic to τ_{c}(V̂′) via the map π|_{τ_{c}(V̂′)}. Let dπ be the derivative of the map π. It is

sufficient to dπ(ξ⃗) ≠ 0 for any non-zero vector ξ⃗ tangent to τ_{c}(V̂′). It is wellknown that the variation vector field of geodesics are Jacobi fields along a geodesic.

Thus, the tangent vectors of τ_{c}(V̂′) at an element (σ̃(t), σ̃′(t)) ∈ τ_{c}(V̂′) are in the form of ξ⃗ = (J(t), J′(t)) for some Jacobi field J along a geodesic σ̃. Hence, dπ((J(t), J′(t)) = J(t). By our construction, τ_{c}(V̂′) is foliated by the geodesics {σ̃} with (σ̃(0), σ̃′(0)) ∈ N^{+}(Ω_{u}, M^{n}). Since Ω_{u} has no focal points in V_{ϵ}(x), for (σ̃(t), σ̃′(t)) ∈ τ_{c}(V̂′), we have dπ(J(t), J′(t)) = J(t) ≠ 0 for any non-trivial tangent vector ξ⃗ = (J(t), J′(t)) of τ_{c}(V̂′) at (σ̃(t), σ̃′(t)). Thus, we conclude that π|_{τ_{c}(V̂′)} is a C^{∞}-smooth diffeomorphism between τ_{c}(V̂′) and V_{ϵ}(x). Therefore, x has a C^{∞}- smooth open neighborhood V_{ϵ}(x) in F_{q}. In summary, we have proved the following fact

Fact 6. (Step 1 in the bootstrap argument) If the subset [U_{δ_{0}(T)}(S) ∩ F_{q}] has the C^{∞}-smooth, k-dimensional interior and if Ω_{u} ⊂ U_{δ_{0}(T)}(S), then the corresponding tubular neighborhood [U_{δ_{0}(T)}(Ω_{u}) ∩ F_{q}] of Ω_{u} (with the same radius) has the C^{∞}- smooth, k-dimensional interior as well. In particular, (5.u) holds for u = u_{1} with 0 ≤ u_{1} ≤ \frac{δ_{0}(T)}{2C_{T}}.

The proof of Fact 6 can be elaborated to verify the following.

Fact 7. (The j-th step in the bootstrap argument) If the subset [U_{δ_{0}(T)}(Ω_{u_{j}}) ∩ F_{q}] has C^{∞}-smooth, k-dimensional interior and if Ω_{u} ⊂ U_{δ_{0}(T)}(Ω_{u_{j}}), then the corresponding tubular neighborhood [U_{δ_{0}(T)}(Ω_{u}) ∩ F_{q}] of Ω_{u} (with the same radius) has C^{∞}-smooth, k-dimensional interior as well. In particular, (5.u) holds for u = u_{j+1} with u_{j} ≤ u_{j+1} ≤ u_{j} + \frac{δ_{0}(T)}{2C_{T}}.

Choose N ≥ 2C_{T} [\frac{T}{δ_{0}(T)} + 1]. Applying the above Fact 7 repeatedly (at most N times), one can complete the proof of (5.u) for u = T. Since M^{n} = ∪_{T} Ω_{T}, Theorem 1 follows from (5.T) for any arbitrarily large T. □

The first author is very grateful to Professor B. Wilking for pointing out a mistake in an earlier version of the manuscript [CaS].

References

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[Wi] Wilking, B., A duality theorem for singular Riemannian foliations in nonnegative sectional curvature, in preparation.

[Y1] Yim, J.-W., Distance nonincreasing retraction on a complete open manifold of nonnegative sectional curvature., Ann. Global Anal. Geom. 6 (1988,), 191–206.

[Y2] , Spaces of souls in a complete open manifold of nonnegative curvature, J. Diff. Geom. 32 (1990,), 429-455.

Jianguo Cao, Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556 USA.

E-mail address: cao.7@nd.edu

Mei-Chi Shaw, Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556 USA

   E-mail address: mei-chi.shaw.1@nd.edu