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