2015/12/13 by Tillmann Jentsch, Jentsch, Tillmann
Mathematics · Medicine · Physics and Astronomy · #53B20 #53C25 #Advanced Differential Geometry Research #Advanced Neuroimaging Techniques and Applications #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.1512.04046
openalex publication_date 2015/12/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Weitzenböck 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.