ABSTRACT
Our ultimate goal is to study abstract surfaces, that is, 2-dimensional manifolds which have a notion of metric compatible with their manifold structure (see Definition 25.21 on page 858). The theory of differentiable manifolds developed in the previous chapter is insufficient on its own to measure lengths and areas. For that we need to impose additional structure so that we end up with a gen-eralization of a metric as defined in Section 12.1. Riemann 1 in his thesis [Riem] (written in 1854 but published only in 1867) introduced such a metric ds 2 in n dimensions and described how the curvature o6f ds 2 can be measured.
