Osserman manifolds

Abstract

Let (M, g) be a pseudo-Riemannian manifold with curvature tensor R. The Jacobi operator RX is the symmetric endomorphism of TpM defined by RX (Y ) = R(Y, X)X. In the Riemannian setting, if M is locally a rank-one symmetric space or if M is flat, then the local isometry group acts transitively on the unit sphere bundle SM . Consequently, the eigenvalues of RX are constant on SM . In the late 1980s, Osserman raised the question of whether the converse also holds; this problem is now known as the Osserman conjecture. In the first part of the lecture, we will present an overview of Osserman-type problems in pseudo-Riemannian geometry, based on the paper: N. Blažić, N. Bokan, Z. Rakić: Osserman Pseudo-Riemannian manifolds of Signature (2, 2), J. Austral. Math. Soc., 71 (2001), 367–395, as well as other related works by the same authors. The second part of the lecture will be devoted to the equivalence between the pointwise Osserman condition and the duality principle. This part is based on joint results obtained in collaboration with Yury Nikolayevsky and Vladica Andrejić. This lecture is dedicated to the memory of my mentor, Professor Novica Blažić, who passed away two decades ago.