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The Jet Isomorphism Theorem of Riemannian Geometry

Tillmann Jentsch

TL;DR

The paper addresses the jet isomorphism problem in Riemannian geometry by constructing a coordinate-free, intrinsic inverse to the jet symmetrization map that recovers the curvature jet $\nabla^{\le k}|_p\mathrm{R}$ from the symmetrized data $\mathcal{R}^{\le k}|_p$. It develops a detailed, elementary proof for the linear (minimal) case of the inverse via Young symmetrizers and the Kulkarni–Nomizu product, and then extends to arbitrary $k$-jets by incorporating Ricci-identity corrections and explicit quadratic terms in lower jets. The framework treats symmetrized curvature jets as a graded module with Jacobi-type relations, enabling algebraic approaches to natural curvature equations and providing concrete, formula-driven reconstructions of $\nabla^{\le k}|_p\mathrm{R}$ from $\mathcal{R}^{\le k}|_p$. The results yield a practical, coordinate-free mechanism to compute the curvature jet in geodesic normal coordinates and connect jet data to Taylor expansions of the metric, with potential PDE applications and broader implications for the structure of curvature constraints. Overall, the work deepens understanding of the jet isomorphism, delivers explicit inversion formulas, and offers a robust algebraic viewpoint for higher-order curvature jets in 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.

The Jet Isomorphism Theorem of Riemannian Geometry

TL;DR

The paper addresses the jet isomorphism problem in Riemannian geometry by constructing a coordinate-free, intrinsic inverse to the jet symmetrization map that recovers the curvature jet from the symmetrized data . It develops a detailed, elementary proof for the linear (minimal) case of the inverse via Young symmetrizers and the Kulkarni–Nomizu product, and then extends to arbitrary -jets by incorporating Ricci-identity corrections and explicit quadratic terms in lower jets. The framework treats symmetrized curvature jets as a graded module with Jacobi-type relations, enabling algebraic approaches to natural curvature equations and providing concrete, formula-driven reconstructions of from . The results yield a practical, coordinate-free mechanism to compute the curvature jet in geodesic normal coordinates and connect jet data to Taylor expansions of the metric, with potential PDE applications and broader implications for the structure of curvature constraints. Overall, the work deepens understanding of the jet isomorphism, delivers explicit inversion formulas, and offers a robust algebraic viewpoint for higher-order curvature jets in 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.

Paper Structure

This paper contains 17 sections, 15 theorems, 127 equations.

Table of Contents

  1. Overview
  2. The inverse of the jet symmetrization map
  3. Outlook: Natural equations for jets of the curvature tensor
  4. Revising the well-known parts of the Jet Isomorphism Theorem
  5. Geodesic normal coordinates
  6. Proof of the jet isomorphism theorem in its usual form
  7. The inverse of the jet symmetrization map R bullet
  8. A direct proof of the Young projection formula (Young Symmetrizer Eq.) for linear k-jets
  9. Proof of \ref{['eq:Young_Symmetrizer']} for $k\ge 2$
  10. Generalization of (Young Symmetrizer Eq.) for arbitrary k-jets
  11. Formulas relating jetk̂ psi and nablak̂ psi
  12. Taylor expansion of the parallel transport
  13. Explicit calculations for the jet formula from Proposition \ref{['p:Simple_Jet-Formula']}
  14. Taylor expansion of the metric in normal coordinates
  15. Taylor expansion of the backward parallel transport
  16. ...and 2 more sections

Key Result

Theorem 1

Let $p \in M$, let $\langle\,\,\cdot\,,\,\cdot\,\,\rangle \coloneqq g|_p$ denote the inner product on $T_pM$, and $\exp^M_p\colon U \to M$ be the exponential map, defined on an open star-shaped neighborhood $U \subset T_pM$ of the origin. Let $\tilde{g}$ denote the pullback of $g$ under $\exp^M_p$, is the Taylor polynomial of order $k$ for the function $X \mapsto \tilde{g}(X)_{Y,Z}$ on $T_pM$ for

Theorems & Definitions (34)

  • Theorem 1: ABP, Gray
  • Definition 1
  • Definition 2
  • Theorem 2: Ep
  • Corollary 1
  • Definition 3: KMS
  • Remark 1
  • Theorem 3
  • proof
  • Remark 2
  • ...and 24 more