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.