Geometric consequences of intrinsic and extrinsic curvature conditions

When studying the geometric properties of a semi–Riemannian manifold, the starting point usually comes from some invariants of the metric structure. Among those invariants, the curvature tensor is perhaps the simplest and most natural one. In the words of R. Osserman:

The notion of curvature is one of the central concepts of differential geometry; one could argue that it is the central one, distinguishing the geometrical core of the subject from those aspects that are analytical, algebraic or topological. In the words of M. Berger, curvature is the “Number 1 Riemannian invariant and the most natural. Gauss and then Riemann saw it instantly”.