The Gauss-Bonnet Theorem
نویسنده
چکیده
منابع مشابه
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.
متن کاملThe 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 کاملA Proof of the Gauss-bonnet Theorem
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 k...
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 - Grotemeyer Theorem in spaces of constant curvature ∗
In 1963, K.P. Grotemeyer proved an interesting variant of the Gauss-Bonnet Theorem. Let M be an oriented closed surface in the Euclidean space R 3 with Euler characteristic χ(M), Gauss curvature G and unit normal vector field n. Grote-meyer's identity replaces the Gauss-Bonnet integrand G by the normal moment (a · n) 2 G, where a is a fixed unit vector: M (a · n) 2 Gdv = 2π 3 χ(M). We generaliz...
متن کاملAnalytic Continuation, the Chern-gauss-bonnet Theorem, and the Euler-lagrange Equations in Lovelock Theory for Indefinite Signature Metrics
We use analytic continuation to derive the Euler-Lagrange equations associated to the Pfaffian in indefinite signature (p, q) directly from the corresponding result in the Riemannian setting. We also use analytic continuation to derive the Chern-Gauss-Bonnet theorem for pseudo-Riemannian manifolds with boundary directly from the corresponding result in the Riemannian setting. Complex metrics on...
متن کامل