The Orthogonality Principle for Osserman Manifolds

Abstract

We introduce a new potential characterization of Osserman algebraic curvature tensors. An algebraic curvature tensor is Jacobi-orthogonal if $\mathcal{J}_XY\perp\mathcal{J}_YX$ holds for all $X\perp Y$, where $\mathcal{J}$ denotes the Jacobi operator. We prove that any Jacobi-orthogonal tensor is Osserman, while all known Osserman tensors are Jacobi-orthogonal.