This book contains a clear exposition of two contemporary topics in modern differential geometry:

• distance geometric analysis on manifolds, in particular, comparison theory for distance functions in spaces which have well defined bounds on their curvature
• the application of the Lichnerowicz formula for Dirac operators to the study of Gromov's invariants to measure the K-theoretic size of a Riemannian manifold.

It is intended for both graduate students and researchers.