Natural curvature operators of bounded spectrum

Let MM be a pseudo-Riemannian manifold of signature (p,q)(p,q) where p⩾1p⩾1 and q⩾1q⩾1. If the Jacobi operator has pointwise bounded spectrum on the pseudo-sphere bundles of unit spacelike or timelike vectors, then MM is pointwise Osserman. Similar results are established for other natural operators of Riemannian geometry. Rigidity phenomena in Lorentzian geometry are discussed.