This book is meant to be an introduction to Riemannian geometry. The reader is assumed to have some knowledge of standard manifold theory, including basic theory of tensors, forms, and Lie groups. At times we shall also assume familiarity with algebraic topology and de Rham cohomology. Specifically, we recommend that the reader is familiar with texts like [14] or[76, vol. 1]. For the readers who have only learned something like the first two chapters of [65], we have an appendix which covers Stokes' theorem, Cech cohomology, and de Rham cohomology. The reader should also have a nodding acquaintance with ordinary differential equations. For this, a text like [59] is more than sufficient. Most of the material usually taught in basic Riemannian geometry, as well as several more advanced topics, is presented in this text. Many of the theorems from Chapters 7 to 11 appear for the first time in textbook form. This is particularly surprising as we have included essentially only the material students ofRiemannian geometry must know. The approach we have taken deviates in some ways from the standard path. First and foremost, we do not discuss variational calculus, which is usually the sine qua non of the subject. Instead, we have taken a more elementary approach that simply uses standard calculus together with some techniques from differential equations
CONTENT
1 Riemannian Metrics -- 2 Curvature -- 3 Examples -- 4 Hypersurfaces -- 5 Geodesics and Distance -- 6 Sectional Curvature Comparison I -- 7 The Bochner Technique -- 8 Symmetric Spaces and Holonomy -- 9 Ricci Curvature Comparison -- 10 Convergence -- 11 Sectional Curvature Comparison II -- A de Rham Cohomology -- A.1 Elementary Properties -- A.2 Integration of Forms -- A.3 ?ech Cohomology -- A.4 de Rham Cohomology -- A.5 Poincarรฉ Duality -- A.6 Degree Theory -- A.7 Further Study -- B Principal Bundles -- B.1 Cartan Formalism -- B.2 The Frame Bundle -- B.3 Construction of the Frame Bundle -- B.4 Construction of Tensor Bundles -- B.5 Tensors -- B.6 The Connection on the Frame Bundle -- B.7 Covariant Differentiation of Tensors -- B.8 Principal Bundles in General -- B.9 Further Study -- C Spinors -- C.1 Spin Structures -- C.2 Spinor Bundles -- C.3 The Weitzenbรถck Formula for Spinors -- C.4 The Square of a Spinor -- C.5 Further Study -- References
