# Critical points of distance functions and applications to geometry

@inproceedings{Cheeger1991CriticalPO, title={Critical points of distance functions and applications to geometry}, author={Jeff Cheeger}, year={1991} }

8. Introduction Critical points of distance functions Toponogov's theorem; first application:a Background on finiteness theorems Homotopy Finiteness Appendix. Some volume estimates Betti numbers and rank Appendix: The generalized Mayer-Vietoris estimate Rank, curvature and diameter Ricci curvature, volume and the Laplacian Appendix. The maximum principle Ricci curvature, diameter growth and finiteness of topological type. Appendix. Nonnegative Ricci curvature outside a compact set.

## 127 Citations

### Applications of critical point theory of distance functions to geometry

- Mathematics
- 1991

In this note, we give two applications of the critical point theory of distance functions to Riemannian geometry. First, we present a new proof of the theorem: if a complete open nonnegatively curved…

### A finite Topological Type Theorem for open manifolds with Non-negative Ricci Curvature and Almost Maximal Local Rewinding Volume

- Mathematics
- 2022

. In this paper, we prove ﬁnite topological type theorems for open manifolds with non-negative Ricci curvature under a new additional regularity assumption on metrics, almost maximal local rewinding…

### Open manifolds with asymptotically nonnegative Ricci curvature and large volume growth

- Mathematics
- 2014

In this paper, we study the topology of complete noncompact Riemannian manifolds with asymptotically nonnegative Ricci curvature and large volume growth. We prove that they have finite topological…

### A Volume Comparison Estimate with Radially Symmetric Ricci Curvature Lower Bound and Its Applications

- MathematicsInt. J. Math. Math. Sci.
- 2010

The classical Bishop-Gromov volume comparison is extended to radially symmetric Ricci curvature lower bound, and applied to investigate the volume growth, total Betti number, and finite topological type of manifolds with nonasymptotically almost nonnegative RicCI curvature.

### Finite Topological Type and Volume Growth

- Mathematics
- 2004

We show that a complete noncompact n-dimensional Riemannian manifold Mwith Ricci curvature RicM ≥ −(n − 1) and conjugateradius conjM ≥ c > 0 has finite topological type, provided that the volume…

### Ricci curvature, conjugate radius and large volume growth

- Mathematics
- 2003

Abstract In this article, we prove that an n-dimensional complete open Riemannian manifold M with Ricci curvature RicM≥−(n−1) is diffeomorphic to a Euclidean n-space if M has positive conjugate…

### Topology of Manifolds with Asymptotically Nonnegative Ricci Curvature

- Mathematics
- 2008

In this paper, we study the topology of complete noncompact Riemannian manifolds with asymptotically nonnegative Ricci curvature. We show that a complete noncompact manifold with asymptoticaly…

### Examples of manifolds of positive Ricci curvature with quadratically nonnegatively curved infinity and infinite topological type.

- Mathematics
- 2017

In this paper, we construct a complete n-dim Riemannian manifold with positive Ricci curvature, quadratically nonnegatively curved infinity and infinite topological type. This gives a negative answer…

### Complete manifolds with asymptotically nonnegative Ricci curvature and weak bounded geometry

- Mathematics
- 2007

Abstract.In this paper, we study complete Riemannian n-manifolds (n ≥ 3) with asymptotically nonnegative Ricci curvature and weak bounded geometry. We show among other things that the total Betti…

## References

SHOWING 1-10 OF 33 REFERENCES

### Curvature, diameter and betti numbers

- Mathematics
- 1981

We give an upper bound for the Betti numbers of a compact Riemannian manifold in terms of its diameter and the lower bound of the sectional curvatures. This estimate in particular shows that most…

### FINITENESS THEOREMS FOR RIEMANNIAN MANIFOLDS.

- Mathematics
- 1970

1. The purpose of this paper is to show that if one puts arbitrary fixed bounds on the size of certain geometrical quantities associated with a riemannian metric, then the set of diffeomorphism…

### Controlled topology in geometry

- Mathematics
- 1989

The purpose of the present note is to announce some finiteness theorems for classes of Riemannian manifolds (cf. A, B and D below). Let ^*$(n) denote the class of closed Riemannian «-manifolds with…

### A generalized sphere theorem

- Mathematics
- 1977

A basic problem in Riemannian geometry is the study of relations between the topological structure and the Riemannian structure of a complete, connected Riemannian manifold M of dimension n > 2. By a…

### Complete Ricci-flat Kähler manifolds of infinite topological type

- Mathematics
- 1989

We display an infinite dimensional family of complete Ricci-flat Kähler manifolds of complex dimension 2, for which the second homology is infinitely generated. These are obtained from the…

### The structure of complete manifolds of nonnegative curvature

- Mathematics
- 1968

0. In this paper we describe some results on the structure of complete manifolds of nonnegative sectional curvature. (We will denote such manifolds by M.) Details and related results will appear…

### Ball covering on manifolds with nonnegative Ricci curvature near infinity

- Mathematics
- 1992

Let M be a complete open Riemannian manifold with nonnegative Ricci curvature outside a compact set B. We show that the following ball covering property (see [LT]) is true provided that the sectional…

### Examples of manifolds of positive Ricci curvature

- Mathematics
- 1989

On presente de nouveaux exemples de varietes de Riemann simplement connexes de dimension ≥7 a courbure de Ricci positive, qui n'admettent pas de metrique de courbure sectionnelle non negative dans…

### Lipschitz convergence of Riemannian manifolds.

- Mathematics
- 1988

Soit #7B-C≡#7B-C(n,Λ,So,Vo) l'ensemble de toutes les varietes de Riemann a n dimensions C ∞ compactes connexes de /courbure sectionnelle/ Vo. On montre que cette classe #7B-C a certaines proprietes…

### Comparison and Finiteness Theorems in Riemannian Geometry

- Mathematics
- 1984

This is a survey article on the above subject. A differentiable manifold admits variety of riemannian structures but we don't know in general what is the most adapted metric to the given…