The Jet Isomorphism Theorem of Riemannian Geometry

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.