In University I have to visit a Topology/Geometry course for my bachelor degree. This course is based on Analysis I/II, so multivariable calculus, Riemann Integrals in multiple dimension and the very basic definitions of metric spaces and convergence. Since I like the subject a lot, I wanted to learn more and currently I am studying the brilliant books by John M. Lee (Topological and smooth manifolds, Riemannian manifolds) since it covers the topics in a more formal way then the course does. On p. 167 in Riemannian manifolds one finds the well-known Gauss-Bonnet Theorem:
If $M$ is a triangulated, compact, oriented, Riemannian $2$-manifold, then $$\int_M K dA = 2\pi \chi(M).$$
So my questions are:
- How should I interpret the integral? Lebesgue or Riemann integral? Since nothing is mentioned in the book, I think it should be a Riemann integral since I do not know how to evaluate an integral over a surface, which brings me to my next question.
- How do I evaluate the lefthandside? I mean, $M$ could be any abstract manifold with no embedding. I mean, in $\mathbb{R}^3$, we work with the formula using the parametrization and so on (which gets also messy if $M$ is complex).
For integration on manifolds, you should refer to Integration of differential forms (Wikipedia).
The type of integral itself (Riemann or Lebesgue) is not so important as at the end the functions being integrated are continuous.