A Gauss-Bonnet Theorem for Asymptotically Conical Manifolds and Manifolds with Conical Singularities

Troyanov, MarcMarcone, Adrien Giuliano2019-01-172019-01-172019-01-17201910.5075/epfl-thesis-9275https://infoscience.epfl.ch/handle/20.500.14299/153515The purpose of this thesis is to provide an intrinsic proof of a Gauss-Bonnet-Chern formula for complete Riemannian manifolds with finitely many conical singularities and asymptotically conical ends. A geometric invariant is associated to the link of both the conical singularities and the asymptotically conical ends and is used to quantify the Gauss-Bonnet defect of such manifolds. This invariant is constructed by contracting powers of a tensor involving the curvature tensor of the link. Moreover this invariant can be written in terms of the total Lipschitz-Killing curvatures of the link. A detailed study of the Lipschitz-Killing curvatures of Riemannian manifolds is presented as well as a complete modern version of Chern's intrinsic proof of the Gauss-Bonnet-Chern Theorem for compact manifolds with boundary.enGauss-Bonnet-ChernLipschitz-Killing CurvaturesAsymptotically Conical EndsConical SingularitiesCohn-Vossen Inequality.A Gauss-Bonnet Theorem for Asymptotically Conical Manifolds and Manifolds with Conical Singularitiesthesis::doctoral thesis