We now embark on the full theory, beginning with the concept of a manifold in differential geometry. The meaning of coordinates and coordinate transformations is carefully explained. The metric and its transformation between coordinate frames is discussed. Riemann normal coordinates are described. The concepts of a tangent space and local flatness are discussed and derived. It is shown how to use the metric to calculate distances, areas and volumes, and to describe submanifolds.
Appendix C: Differential Geometry and Coordinate Transformations
