Geometric consequences of algebraic conditions on curvature operators

As a Riemannian invariant, the curvature and its derivatives are the most natural algebraic invariants which stem from the connection. Therefore, this suggests that the curvature encodes a lot of information of the geometry of a Riemannian manifold. These considerations show that the curvature is a fundamental concept in diferential geometry, nevertheless the role played by this important tensor is not yet completely understood. The main purpose of this thesis is to obtain geometric consequences from algebraic conditions on the curvature tensor. Usually, we will impose these conditions on operators associated to the curvature tensor, since the curvature tensor itself is hard to handle. Generally we work in the broad setting of pseudo-Riemannian manifolds; however, in some chapters or sections we will restrict our analysis to positive definite metrics.