Differential Geometry and Homogeneous Spaces
Description
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.
Reviews / Votes
From the reviews of the German language editions:
"This book is unique in modern differential geometry by its way of writing, by the multiple examples and exercises. ... We recommend this book to both undergraduates and PhD-students." (Petre Stavre, zbMATH 1558.53002, 2025)
"It is a very recommended text for these topics. Some classical books in the area are assumed as inspiration for the present text. It is important to notice that maybe no one of those classicals contains all of the information given here." (Gabriela Paola Ovando, zbMATH 1476.53001, 2022)
"Due to its structure the book is aimed at an undergraduate audience; however the detailed description of concepts makes it accessible to established researchers too who are approaching this field." (Corina Mohorianu, zbMATH 1306.53002, 2015)
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Person
Kai Köhler is Professor of Mathematics at the Heinrich Heine University of Düsseldorf. His research area is Geometry, with an emphasis on Global Analysis and Arithmetic Algebraic Geometry.
Content
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.