03369nam a22005415i 4500 978-3-662-69721-4 DE-He213 20251030113250.0 cr nn 008mamaa 241029s2024 gw | s |||| 0|eng d 9783662697214 978-3-662-69721-4 10.1007/978-3-662-69721-4 doi QA641-670 PBMP bicssc MAT012030 bisacsh PBMP thema Köhler, Kai. author. aut http://id.loc.gov/vocabulary/relators/aut 86797 Differential Geometry and Homogeneous Spaces [electronic resource] / by Kai Köhler. 1st ed. 2024. Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2024. X, 292 p. online resource. text txt rdacontent computer c rdamedia online resource cr rdacarrier text file PDF rda Universitext, 2191-6675 1 Manifolds -- 2 Vector Bundles and Tensors -- 3 Riemannian Manifolds -- 4 The Poincaré–Hopf Theorem and the Chern–Gauß–Bonnet Theorem -- 5 Geodesics -- 6 Homogeneous Spaces -- 7 Symmetric Spaces -- 8 General Relativity -- A Solutions to Selected Exercises. This textbook offers a rigorous introduction to the foundations of Riemannian Geometry, with a detailed treatment of homogeneous and symmetric spaces, as well as the foundations of the General Theory of Relativity. Starting with the basics of manifolds, it presents key objects of differential geometry, such as Lie groups, vector bundles, and de Rham cohomology, with full mathematical details. Next, the fundamental concepts of Riemannian geometry are introduced, paving the way for the study of homogeneous and symmetric spaces. As an early application, a version of the Poincaré–Hopf and Chern–Gauss–Bonnet Theorems is derived. The final chapter provides an axiomatic deduction of the fundamental equations of the General Theory of Relativity as another important application. Throughout, the theory is illustrated with color figures to promote intuitive understanding, and over 200 exercises are provided (many with solutions) to help master the material. The book is designed to cover a two-semester graduate course for students in mathematics or theoretical physics and can also be used for advanced undergraduate courses. It assumes a solid understanding of multivariable calculus and linear algebra. Geometry, Differential. 69684 Topological groups. 72299 Lie groups. 72300 Mathematical physics. Differential Geometry. 69686 Topological Groups and Lie Groups. 72304 Mathematical Physics. SpringerLink (Online service) Springer Nature eBook Printed edition: 9783662697207 Printed edition: 9783662697221 Universitext, 2191-6675 73095 https://doi.org/10.1007/978-3-662-69721-4 ZDB-2-SMA ZDB-2-SXMS 64588 64588