A Proof of the Gauss-bonnet Theorem
نویسنده
چکیده
In this paper I will provide a proof of the Gauss-Bonnet Theorem. I will start by briefly explaining regular surfaces and move on to the first and second fundamental forms. I will then discuss Gaussian curvature and geodesics. Finally, I will move on to the theorem itself, giving both a local and a global version of the Gauss-Bonnet theorem. For this paper, I will assume that the reader has a knowledge of point-set topology, analysis in Rn, and linear algebra.
منابع مشابه
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...
متن کاملThe 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.
متن کاملIntegral Geometry and the Gauss-bonnet Theorem in Constant Curvature Spaces
We give an integral-geometric proof of the Gauss-Bonnet theorem for hypersurfaces in constant curvature spaces. As a tool, we obtain variation formulas in integral geometry with interest in its own.
متن کاملThe Gauss-Bonnet-Chern Theorem on Riemannian Manifolds
This expository paper contains a detailed introduction to some important works concerning the Gauss-Bonnet-Chern theorem. The study of this theorem has a long history dating back to Gauss’s Theorema Egregium (Latin: Remarkable Theorem) and culminated in Chern’s groundbreaking work [14] in 1944, which is a deep and wonderful application of Elie Cartan’s formalism. The idea and tools in [14] have...
متن کاملA Renormalized Index Theorem for Some Complete Asymptotically Regular Metrics: the Gauss-bonnet Theorem
The Gauss-Bonnet Theorem is studied for edge metrics as a renormalized index theorem. These metrics include the Poincaré-Einstein metrics of the AdS/CFT correspondence. Renormalization is used to make sense of the curvature integral and the dimensions of the L-cohomology spaces as well as to carry out the heat equation proof of the index theorem. For conformally compact metrics even mod x, the ...
متن کامل