Chapter3 3 3. An Introduction to the Generalized Gauss-Bonnet-Chern Theorem

Abstract This work studies the mathematical structures which are relevant to differentiable manifolds needed to prove the Gauss-Bonnet-Chern theorem. These structures include de Rham cohomology vector spaces of the manifold, characteristic classes such as the Euler class, pfaffians, and some fiber bundles with useful properties. The paper presents a unified approach that makes use of fiber bundles and leads to a non-computational proof of the Gauss-Bonnet-Chern Theorem. It is indicated how it can be generalized to manifolds with boundary.