Introduction to Differential Geometry

Description

This textbook is suitable for a one semester lecture course on differential geometry for students of mathematics or STEM disciplines with a working knowledge of analysis, linear algebra, complex analysis, and point set topology. The book treats the subject both from an extrinsic and an intrinsic view point. The first chapters give a historical overview of the field and contain an introduction to basic concepts such as manifolds and smooth maps, vector fields and flows, and Lie groups, leading up to the theorem of Frobenius. Subsequent chapters deal with the Levi-Civita connection, geodesics, the Riemann curvature tensor, a proof of the Cartan-Ambrose-Hicks theorem, as well as applications to flat spaces, symmetric spaces, and constant curvature manifolds. Also included are sections about manifolds with nonpositive sectional curvature, the Ricci tensor, the scalar curvature, and the Weyl tensor. An additional chapter goes beyond the scope of a one semester lecture course and deals with subjects such as conjugate points and the Morse index, the injectivity radius, the group of isometries and the Myers-Steenrod theorem, and Donaldson's differential geometric approach to Lie algebra theory. Preface Contents 1 What Is Differential Geometry? 1.1 Cartography and Differential Geometry 1.2 Coordinates 1.3 Topological Manifolds* 1.4 Smooth Manifolds Defined* 1.5 The Master Plan 2 Foundations 2.1 Submanifolds of Euclidean Space 2.2 Tangent Spaces and Derivatives 2.3 Submanifolds and Embeddings 2.4 Vector Fields and Flows 2.5 Lie Groups 2.6 Vector Bundles and Submersions 2.7 The Theorem of Frobenius 2.8 The Intrinsic Definition of a Manifold* 2.9 Consequences of Paracompactness* 3 The Levi-Civita Connection 3.1 Second Fundamental Form 3.2 Covariant Derivative 3.3 Parallel Transport 3.4 The Frame Bundle 3.5 Motions and Developments 3.6 Christoffel Symbols 3.7 Riemannian Metrics* 4 Geodesics 4.1 Length and Energy 4.2 Distance 4.3 The Exponential Map 4.4 Minimal Geodesics 4.5 Convex Neighborhoods 4.6 Completeness and Hopf–Rinow 4.7 Geodesics in the Intrinsic Setting* 5 Curvature 5.1 Isometries 5.2 The Riemann Curvature Tensor 5.3 Generalized Theorema Egregium 5.4 Curvature in Local Coordinates* 6 Geometry and Topology 6.1 The Cartan–Ambrose–Hicks Theorem 6.2 Flat Spaces 6.3 Symmetric Spaces 6.4 Constant Curvature 6.5 Nonpositive Sectional Curvature 6.6 Positive Ricci Curvature* 6.7 Scalar Curvature* 6.8 The Weyl Tensor* 7 Topics in Geometry 7.1 Conjugate Points and the Morse Index* 7.2 The Injectivity Radius* 7.3 The Group of Isometries* 7.4 Isometries of Compact Lie Groups* 7.5 Convex Functions on Hadamard Manifolds* 7.6 Semisimple Lie Algebras* Appendix A: Notes A.1 Maps and Functions A.2 Normal Forms A.3 Euclidean Spaces References Index