Puffini-Videv Models and Manifolds

Let \(J(\pi)\) be the higher order Jacobi operator. We study algebraic curvature tensors where \(J(\pi)J(\pi^{\perp})=J(\pi^{\perp})J(\pi)\). In the Riemannian setting, we give a complete characterization of such tensors; in the pseudo-Riemannian setting, partial results are available. We present non-trivial geometric examples of Riemannian manifolds with this property.