The Jet Isomorphism Theorem of Riemannian Geometry

TL;DR

This paper revisits the classical jet isomorphism theorem in Riemannian geometry, providing a coordinate-free proof that simplifies the reconstruction of curvature jets from symmetrized derivatives.

Contribution

It offers a new intrinsic, coordinate-free proof of the jet isomorphism theorem using Young symmetrizers, simplifying the understanding of curvature tensor reconstruction.

Findings

Provides a coordinate-free proof of the jet isomorphism theorem.

Simplifies the process of reconstructing curvature jets from symmetrized derivatives.

Uses intrinsic definitions and Young symmetrizers for clarity and elegance.

Abstract

A classical theorem of Riemannian geometry, due in its original form to Cartan, states that the Taylor expansion of the metric in geodesic normal coordinates is a universal formal power series involving only the symmetrizations of the iterated covariant derivatives of the curvature tensor; this is known as the jet isomorphism theorem. In particular, it is in principle possible to reconstruct the jet of the curvature tensor from its symmetrization in geodesic normal coordinates, although this would certainly result in an unwieldy computation. In this paper we achieve the same goal by coordinate-free calculations, using only the intrinsic definition of the relevant Young symmetrizers.