The two-jet of the curvature tensor of an Einstein manifold

TL;DR

This paper characterizes the two-jet of the curvature tensor on Einstein manifolds using symmetrization conditions, linking geometric properties to algebraic conditions via the Jet Isomorphism Theorem and Weitzenböck formula.

Contribution

It provides necessary and sufficient conditions for the Einstein property of two-jets of curvature tensors in terms of symmetrization and Jacobi operators, extending previous geometric characterizations.

Findings

Derived conditions for Einstein two-jets using symmetrization.

Connected the Einstein property to the Jacobi operator and its derivatives.

Analyzed linear Jacobi relations of order two on Einstein manifolds.

Abstract

The two-jet of the curvature tensor at some point of a pseudo-Riemannian manifold is called Einstein if the Ricci tensor is a multiple of the metric tensor at the given point and additionally its first two covariant derivatives vanish there. Following the Jet Isomorphism Theorem of pseudo-Riemannian geometry, we derive necessary and sufficient conditions for the Einstein property in terms of the symmetrization of the given two-jet (i.e. in terms of the Jacobi operator and its first two covariant derivatives along arbitrary geodesics emanating from the given point). A central role is played by the Weitzenb\"ock formula for the Laplacian d delta + delta d acting on sections of the vector bundle of algebraic curvature tensors. As an application, we study linear Jacobi relations of order two on Einstein manifolds.