ABSTRACT
This chapter introduces the metric as an additional structure, whose central role is to define the geometry of a manifold. It considers the conditions of isometry and conformality between different geometric manifolds. The chapter provides several examples of classical geometries that are relevant for later purposes. It develops the rudiments of integration on manifolds. The chapter introduces differential forms and the exterior derivative so that it can define integration measures for curves, hypersurfaces, and volumes in a unified way. It discusses the integral theorem of Stokes, the Gauss divergence theorem, and a special version of Stokes' theorem for antisymmetric tensor fields. For the ancient Greek geometers, this was understood as a statement about geometric objects, in this case about the squares. At that time, the Pythagorean theorem was not formulated algebraically, nor was it primarily a statement about numbers.
