A Topological Gauss-bonnet Theorem

The generalized Gauss-Bonnet theorem of Allendoerfer-Weil [1] and Chern [2] has played an important role in the development of the relationship between modern differential geometry and algebraic topology, providing in particular one of the primary stimuli for the theory of characteristic classes. There are now a number of proofs in the literature, from the quite sophisticated (deducing it as a special case of the Atiyah-Singer index theorem for example) to the relatively elementary and straightforward. (For a particularly elegant example of the latter see [7, Appendix C].) In general these previous proofs have a definite cohomological flavor and invoke explicit appeals to general vector bundle or principal bundle theory. In view of the above historical fact this is perhaps natural, and yet from another point of view it is somewhat anomalous. For the theorem states the equality of two quantities: