E-Books →Geometry Smooth Manifolds, Pseudo–Riemannian Geometry, Osserman Manifolds

The curvature is the most natural and most important invariant of pseudo-Riemannian geometry. According to Osserman, the notion of curvature is one of the central concepts of differential geometry, distinguishing the geometric core of the subject from those aspects that are analytic, algebraic, or topological. The curvature information is contained in the curvature tensor, which is diffi cult to work with, despite the many symmetriesit possesses. Extracting the geometrical information that is encoded therein is often quite a challenging task. That is why Gromov described the curvature tensor as a little monster of (multi)linear algebra whose full geometric meaning remains obscure. Therefore, instead of working with the curvature tensor itself, we often use Jacobi operators or sectional curvature that are easier to handle and have a better geometric interpretation, while they contain the complete curvature information.
Our general goal is to fi nd some kind of bridge between the curvature of a pseudo-Riemannian manifold and its geometric properties.