Equivalence between Jacobi-orthogonality and Osserman condition in dimension four

Abstract

An algebraic curvature tensor on a (possibly indefinite) scalar product space is said to be Jacobi-orthogonal if, for any mutually orthogonal vectors X and Y, the Jacobi operator of X applied to Y is orthogonal to the Jacobi operator of Y applied to X. We prove that any four-dimensional algebraic curvature tensor is Jacobi-orthogonal if and only if it is Osserman.