The Orthogonality Principle for Osserman Manifolds

We introduce a new potential characterization of Osserman algebraic curvature tensors. An algebraic curvature tensor is Jacobi-orthogonal if ${\mathcal{J}}_{X}Y\perp {\mathcal{J}}_{Y}X$ 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.