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 study of scalar curvature rigidity and positive mass theorems using spinors and the Dirac operator

It is intended for both graduate students and researchers.



This second edition has been updated to include recent developments such as promising results concerning the geometry of exit time moment spectra and potential analysis in weighted Riemannian manifolds, as well as results pertaining to an early conjecture on the geometry of the scalar curvature and speculations on new geometric approaches to the Index Theorem.