Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Distance Functions, Curvature and Topology

Carlo Mantegazza, Francesca Oronzio

Abstract

We discuss some properties of the distance functions on Riemannian manifolds and we relate their behavior to the geometry of the manifolds. This leads to alternative proofs of some "classical" theorems connecting curvature and topology.

Distance Functions, Curvature and Topology

Abstract

We discuss some properties of the distance functions on Riemannian manifolds and we relate their behavior to the geometry of the manifolds. This leads to alternative proofs of some "classical" theorems connecting curvature and topology.
Paper Structure (4 sections, 13 theorems, 99 equations)

This paper contains 4 sections, 13 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Distance functions
  3. The distance function from a point
  4. Applications

Key Result

Lemma 2.2

Let $r:U\rightarrow {{\mathbb R}}$ be a distance function. Then we have Hence, $\mathrm{S}(\partial_r)=0$, that is, $\partial_r$ is a null eigenvector of $\mathrm{S}$ and $\mathrm{Hess}\,r(\partial_r,\cdot)=0$.

Theorems & Definitions (27)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4: Curvature equations
  • Proposition 2.5: Equations of Riemannian geometry
  • Corollary 2.6
  • Definition 2.7
  • Proposition 2.8
  • Proposition 2.9
  • ...and 17 more