On Osserman condition in pseudo-Riemannian geometry

Let $(M,g)$ be a pseudo-Riemannian manifold, with curvature tensor $R$. The Jacobi operator $R_{X}$ is the symmetric endomorphism of $T_{p}M$ defined by $R_{X}(Y)=R(Y,X)X.$ In Riemannian settings, 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 $SM$ and hence the eigenvalues of ${R} _{X}$ are constant on $SM$. Osserman in the late eighties, wondered if the converse held; this question is usually known as the Osserman conjecture. In the last twenty years many authors have been studied problems which arising from the Osserman conjecture and its generalizations. In the first part of the lecture we will give an overview of Osserman type problems in the pseudo-Riemannian geometry. The second part is devoted to the equivalence of the Osserman pointwise condition and the duality principle. This part of the lecture consists the new results obtained in collaboration with Yury Nikolayevsky.