Comparison Theory of Distance Spheres along Geodesics

Abstract

Estimates for the principal curvature of distance spheres in Riemannian manifolds with sectional curvature bounded from below are well known. The same holds for the mean curvature of distance spheres in Riemannian manifolds with Ricci curvature bounded from below.

In this article we present new estimates for convexity properties of the distance function of a point under different assumptions, for example for manifolds with lower bounds on the conjugate or on the focal radius in addition to these curvature conditions.

The main idea is to introduce a new tensor field describing the differential of the exponential map and verifying a Riccati equation. This technique allows us to get new estimates for the volume form and for Jacobi fields in this context but also to gain new insights into well-known comparison theorems.