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

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.