ABSTRACT
To recapitulate what we have done in the past two chapters, manifolds are topological spaces that are locally homeomorphic to a Euclidean space Rn in which one could do calculus. Chapter 4 introduced the analysis on manifolds by connecting it to analysis on Rn via coordinate charts. However, the astute reader might have noticed that our presentation of analysis on manifolds so far has not recovered one of the foundational aspects of Euclidean calculus: the concept of distance. And related to the concept of distance are angles, areas, volumes, curvature, etc.
