Jacobi–Jacobi Commuting Models and Manifolds

Abstract

Let \({\mathcal{J}}(\pi)\) be the higher order Jacobi operator. We study algebraic curvature tensors where \({\mathcal{J}} (\pi){\mathcal{J}} (\pi^{\bot}) = {\mathcal{J}} (\pi^{\bot}){\mathcal{J}} (\pi)\). 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.