Infinitesimal isometries along curves and generalized Jacobi equations

TL;DR

This paper extends the concept of Jacobi fields to arbitrary curves on Riemannian manifolds, introducing infinitesimal isometries along curves and analyzing their properties through Killing transport and related connections.

Contribution

It generalizes Jacobi fields to arbitrary curves and develops methods to compute and analyze infinitesimal isometries along these curves.

Findings

Killing transport is parallel transport for a specific connection on the jet bundle.

Curvature of this connection relates to local obstructions to infinitesimal isometries in dimension two.

Two approaches to defining infinitesimal isometries are provided: prolongation of the Killing equation and variational equations.

Abstract

A Jacobi field on a Riemannian manifold M is defined along a geodesic. We generalize this notion to an arbitrary smooth curve, and call it an infinitesimal isometry along the curve. We give two approaches to this: 1) compute the complete prolongation of the Killing equation and then restrict to the curve, and 2) compute the variational equations of a rigid motion of the curve. This results in Killing transport along the curve, which is parallel transport for a related connection on the jet bundle J(TM). We study the curvature and holonomy of this connection. In particular, in dimension two the curvature is the local obstruction to infinitesimal isometries on M.