Abstract:
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.
