The metric projection onto the soul

Abstract

Abstract. We study geometric properties of the metric projection π : M → S of an open manifold M with nonnegative sectional curvature onto a soul S. π is shown to be C ∞ up to codimension 3. In arbitrary codimensions, small metric balls around a soul turn out to be convex, so that the unit normal bundle of S also admits a metric of nonnegative curvature. Next we examine how the horizontal curvatures at infinity determine the geometry of M , and study the structure of Sharafutdinov lines. We conclude with regularity properties of the cut and conjugate loci of M .The resolution of the Soul conjecture of Cheeger and Gromoll by Perelman showed that the structure of open manifolds with nonnegative sectional curvature is more rigid than expected. One of the key results in [14] is that the metric projection π : M → S which maps a point p in M to the point π(p) in S that is closest to p is a Riemannian submersion. Perelman also observed that this map is at least of class C 1 , and later it was shown in [9] that π is at least C 2 , and C ∞ at almost every point.The existence of such a Riemannian submersion combined with the restriction on the sign of the curvature suggests that the geometry of M must be special. The purpose of this paper is to illustrate this in several different directions. In fact, we show that most natural geometric objects classically used in the study of these spaces are intrinsically related to the structure of the map π.The paper is essentially structured as follows: After introducing notation and recalling some basic results in section 1, we establish in section 2 the existence of maps similar to π for any submanifold of M homologous and isometric to the soul, and use this to prove the existence of convex tubular neighborhoods for them. In particular, this also proves: Theorem 2.5. The unit normal bundle ν 1 (S) admits a metric with nonnegative sectional curvature.As a consequence, open manifolds with nonnegative curvature provide examples of compact manifolds with the same lower curvature bound (see also [8]). Theorem 2.5 also implies that nontrivial plane bundles over a Bieberbach manifold do not admit nonnegatively curved metrics, thereby providing a shorter proof of the main result in [13]. We are grateful to the referee for pointing out this fact to us.Properties such as these are established via standard submersion techniques, but the fact that π is not known to be smooth everywhere prevents us from using these techniques to study the geometry of M far away from the soul. One way around

Bounded Riemannian submersions

“…The second consequence of compact holonomy relates to the holonomy diffeomorphisms. It is straightforward to show that the holonomy diffeomorphism h α satisfies the Lipschitz constant e CT •length(α) ; see [4,Lemma 4.2]. We prove next that, in the case of compact holonomy, the holonomy diffeomorphisms satisfy a Lipschitz bound which does not depend on the length of the path.…”

“…Guijarro and Walschap asked whether, for the metric projection onto a soul, all holonomy diffeomorphisms must satisfy a global Lipschitz bound [4]. The reason for their interest in this question will be discussed in section 4.…”

Untitled

“…In particular, in Section 6 we establish analogs of Theorem 1.3 for C's that belong to several classes of sphere bundles. Note that sphere bundles over closed nonnegatively curved manifolds are potentially a good source of compact manifolds with sec ≥ 0, because the unit sphere bundle of the normal bundle to the soul is nonnegatively curved [GW00]. Also in Section 8, we prove an analog of Theorem 1.3 where C is any currently known simply-connected positively curved manifold.…”

Топология Слоений Коразмерности 1 Неотрицательной Кривизны

“…[27]). Пусть p, q ∈ S и α : [0, l] → S -минималь-ная геодезическая, соединяющая p и q. Если h : F p → F q -соответствующий диффеоморфизм голономии, который определяется горизонтальными подня-тиями α, то h липшицево.…”