Gauss-Bonnet theorem for 2-dimensional foliations

The Gauss-Bonnet Theorem

The Gauss-bonnet Theorem

The Gauss Bonnet theorem links differential geometry with topology. The following expository piece presents a proof of this theorem, building up all of the necessary topological tools. Important applications of this theorem are discussed.

Visualization of Gauss-Bonnet Theorem

The sum of external angles of a polygon is always constant, π 2 . There are several elementary proofs of this fact. In the similar way, there is an invariant in polyhedron that is π 4 . To see this, let us consider a regular tetrahedron as an example. Tetrahedron has four vertices. Three regular triangles gather at each vertex. Developing the tetrahedron around each vertex, there is an open ang...

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 ...

The Gauss-Bonnet Theorem for Vector Bundles

We give a short proof of the Gauss-Bonnet theorem for a real oriented Riemannian vector bundle E of even rank over a closed compact orientable manifold M . This theorem reduces to the classical Gauss-Bonnet-Chern theorem in the special case when M is a Riemannian manifold and E is the tangent bundle of M endowed with the Levi-Civita connection. The proof is based on an explicit geometric constr...