Non-negatively curved manifolds which are flat outside a compact set

Greene, R. E.; Wu, H.

doi:10.1090/pspum/054.3/1216628

Cited by 3 publications

(7 citation statements)

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“…As motivation for the above question, recall from [5] that if M is flat outside a compact set, then it splits locally isometrically over S. This can be generalized as follows to vertizontal planes, i.e., planes spanned by a vertical and a horizontal vector. …”

Section: Proof Let γ : [0 1] → M Be a Minimal Geodesic Joining π(P)mentioning

confidence: 99%

The metric projection onto the soul

Guijarro

Walschap

1999

Trans. Amer. Math. Soc.

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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

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Section: Proof Let γ : [0 1] → M Be a Minimal Geodesic Joining π(P)mentioning

confidence: 99%

The metric projection onto the soul

Guijarro

Walschap

1999

Trans. Amer. Math. Soc.

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“…By (5), n is in η ⊥ . (7) means that the rank of Ric restricted to the hypersurfaces M t for t > 0 is at most 1. Finally, the meancurvature H t = R − Ric(n, n) = Ric(n, n) is positive, unless it would contradict (5).…”

Section: The Global Splittingmentioning

confidence: 99%

“…Then the result is that the curvature decays exponentially. In this way, let us mention a result of Greene and Wu [7] on nonnegatively curved spaces which are flat at infinity. Theorem 1.3 (Greene-Wu).…”

Section: Introductionmentioning

confidence: 99%

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Steady gradient Ricci soliton with curvature in $L^1$

Deruelle

2012

Communications in Analysis and Geometry

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“…In several independent papers, it has been shown that results concerning topological finiteness and rigidity of open manifolds with nonnegative sectional curvature can be extended to manifolds with nonnegative sectional curvature outside a compact set (see for instance [4], [20], [29]), [30]); the next step was to ask if those results would still hold for manifolds of nonnegative curvature at infinity (and how to define this notion). In case the curvatures decay faster then quadratically, one possible way to handle this problem, can be found in the paper "Lower curvature bounds, Toponogov's Theorem, and bounded topology" of U. Abresch [1], where he studied manifolds of asymptotically nonnegative curvature, which generalize in an interesting way the notion of a manifold with nonnegative sectional curvature at infinity; also a generalized Toponogov Comparison Theorem is proved for this class of manifolds.…”

Section: A Introductionmentioning

confidence: 99%

Manifolds with asymptotically nonnegative minimal radial curvature

Santos

2007

Advg

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In this paper we extend results on the geometry of manifolds with asymptotically nonnegative curvature to manifolds with asymptotically nonnegative minimal radial curvature, showing that most of the results obtained by U. Abresh [1], A. Kasue [28] and S. Zhu [48] hold in a more general context. Particularly, we show that there exists one and only one tangent cone at infinity to each such manifold, in contrast with the class of manifolds of nonnegative Ricci curvature. A. IntroductionIn several independent papers, it has been shown that results concerning topological finiteness and rigidity of open manifolds with nonnegative sectional curvature can be extended to manifolds with nonnegative sectional curvature outside a compact set (see for instance [4], [20], [29]), [30]); the next step was to ask if those results would still hold for manifolds of nonnegative curvature at infinity (and how to define this notion). In case the curvatures decay faster then quadratically, one possible way to handle this problem, can be found in the paper "Lower curvature bounds, Toponogov's Theorem, and bounded topology" of U. Abresch [1], where he studied manifolds of asymptotically nonnegative curvature, which generalize in an interesting way the notion of a manifold with nonnegative sectional curvature at infinity; also a generalized Toponogov Comparison Theorem is proved for this class of manifolds. After Abresch's paper, some results were obtained on the geometry of this class of manifolds, for instance: in [28], there is a comprehensive discussion on the ideal boundaries of this class of manifolds; in [48], a volume comparison theorem is proved. On the other hand, a careful reading of Abresch, Kasue and Zhu * supported by CNPq, UNICAMP-Brazil. 332Newton L. Santos results shows that the condition related to the monotonicity of the sectional curvature can be reduced, without loss of generality, to a condition on the monotonicity of the minimal radial curvatures, because a substantial part of the information one can obtain on the geometry of a manifold with base point (M n , o) is given by the behavior of the geodesic spray, from o (that is, a substantial amount of information of the manifold is contained in the behavior of its radial Jacobi fields). For instance, the Bonnet-Myers Theorem and the Bishop-Gromov Comparison Theorem hold for manifolds with minimal radial curvature bounded below. Also, the Rauch and Berger Comparison Theorems admit a version for minimal radial curvature Most of the results which will be developed next are, actually, obvious reformulations of results presented in Abresch [1], Kasue [28] and Zhu [48], that is, the following results have already been developed and proved by the above mentioned mathematicians, in a context of less generality.Recall that the minimal radial curvatures K min o (p) at a point p in M n are just the sectional curvatures restricted to planes tangent to minimizing geodesics departing from a base point o in M n . A.1. Definition. A complete open Riemannian manifold (M n , o, g) wit...

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Non-negatively curved manifolds which are flat outside a compact set

Cited by 3 publications

References 0 publications

The metric projection onto the soul

The metric projection onto the soul

Steady gradient Ricci soliton with curvature in $L^1$

Manifolds with asymptotically nonnegative minimal radial curvature

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