Notes on the Jacobi equation

TL;DR

This paper explores properties of Jacobi fields in Riemannian geometry without assuming curvature conditions, comparing indices of various Jacobi field spaces and providing applications to geometric analysis.

Contribution

It introduces new insights into Jacobi fields by analyzing their properties without curvature assumptions and compares indices across different spaces, with applications.

Findings

Comparison of indices of Jacobi field spaces

Properties of Jacobi fields independent of curvature assumptions

Applications to Riemannian geometry

Abstract

We discuss some properties of Jacobi fields that do not involve assumptions on the curvature endomorphism. We compare indices of different spaces of Jacobi fields and give some applications to Riemannian geometry.