The Lichnerowicz Formula and Lower Bounds for the Scalar Curvature

This is a short essay describing some fundamental results about rigidity and other consequences of lower bounds on the scalar curvature which are obtained by using the Lichnerowicz formula and its variants. We first discuss positive mass theorems and their hyperbolic analogue on non-compact manifolds. These results are established without using the Atiyah-Singer Index Theorem. In the next section we discuss results about K-Area and K-length on compact manifolds where the use of the Index Theorem is essential. We end with some speculative remarks.