Riemannian Manifolds: An Introduction to Curvature (Graduate Texts in Mathematics (176))

This text is designed for a one-quarter or one-semester graduate couse in Riemannian geometry. It focuses on developing an intimate acquaintance with the geometric meaning of curvature and thereby introduces and demonstrates all the main technical tools needed for a more advanced course on Riemannian manifolds. The book begins with a careful treatment of the machinery of metrics, connections, and geodesics, and then introduces the Riemann curvature tensor, before moving on the submanifold theory, in order to give the curvature tensor a concrete quantitative interpretation. The remainder of the text is devoted to proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet's Theorem, and a special case of the Cartan-Ambrose- Hicks Theorem. This unique volume will especially appeal to students by presenting a selective introduction to the main ides of the subject in an easily accessible way. The material is ideal for a single course, but broad enough to provide students with a firm foundation from which to pursue research or develop applications in Riemannian geometry and other fields that use its tools. Of special interest are the exercises and problems dispersed throughout the text. The exercises are carefully chosen and timed so as to give the reader opportunities to review material that hasjust been introduced, to practice working with the definitions, and to develop skills that are used later in the book. The problems that conclude the chapters are generally more difficult. They not only introduce new mateiral not covered in the body of the text, but they also provide the students with indispensable practice in using the Cover......Page 1 Series: Graduate Texts in Mathematics 176......Page 2 Riemannian Manifolds: An Introduction to Curvature......Page 4 Copyright......Page 5 Preface......Page 8 Contents......Page 14 1. What Is Curvature?......Page 18 The Euclidean Plane......Page 19 Surfaces in Space......Page 21 Curvature in Higher Dimensions......Page 25 Tensors on a Vector Space......Page 28 Manifolds......Page 31 Vector Bundles......Page 33 Tensor Bundles and Tensor Fields......Page 36 Riemannian Metrics......Page 40 Elementary Constructions Associated with Riemannian Metrics......Page 44 Generalizations of Riemannian Metrics......Page 47 The Model Spaces of Riemannian Geometry......Page 50 Problems......Page 60 4. Connections......Page 64 The Problem of Differentiating Vector Fields......Page 65 Connections......Page 66 Vector Fields Along Curves......Page 72 Geodesics......Page 75 Problems......Page 80 The Riemannian Connection......Page 82 The Exponential Map......Page 89 Normal Neighborhoods and Normal Coordinates......Page 93 Geodesics of the Model Spaces......Page 98 Problems......Page 104 Lengths and Distances on Riemannian Manifolds......Page 108 Geodesics and Minimizing Curves......Page 113 Completeness......Page 125 Problems......Page 129 Local Invariants......Page 132 Flat Manifolds......Page 136 Symmetries of the Curvature Tensor......Page 138 Ricci and Scalar Curvatures......Page 141 Problems......Page 145 8. Riemannian Submanifolds......Page 148 Riemannian Submanifolds and the Second Fundamental Form......Page 149 Hypersurfaces in Euclidean Space......Page 156 Geometric Interpretation of Curvature in Higher Dimensions......Page 162 Problems......Page 167 9. The Gauss–Bonnet Theorem......Page 172 Some Plane Geometry......Page 173 The Gauss–Bonnet Formula......Page 179 The Gauss–Bonnet Theorem......Page 183 Problems......Page 188 10. Jacobi Fields......Page 190 The Jacobi Equation......Page 191 Computations of Jacobi Fields......Page 195 Conjugate Points......Page 198 The Second Variation Formula......Page 202 Geodesics Do Not Minimize Past Conjugate Points......Page 204 Problems......Page 208 11. Curvature and Topology......Page 210 Some Comparison Theorems......Page 211 Manifolds of Negative Curvature......Page 213 Manifolds of Positive Curvature......Page 216 Manifolds of Constant Curvature......Page 221 Problems......Page 225 References......Page 226 Index......Page 230 This book is designed as a textbook for a one-quarter or one-semester graduate course on Riemannian geometry, for students who are familiar with topological and differentiable manifolds. It focuses on developing an intimate acquaintance with the geometric meaning of curvature. In so doing, it introduces and demonstrates the uses of all the main technical tools needed for a careful study of Riemannian manifolds. The author has selected a set of topics that can reasonably be covered in ten to fifteen weeks, instead of making any attempt to provide an encyclopedic treatment of the subject. The book begins with a careful treatment of the machinery of metrics, connections, and geodesics,without which one cannot claim to be doing Riemannian geometry. It then introduces the Riemann curvature tensor, and quickly moves on to submanifold theory in order to give the curvature tensor a concrete quantitative interpretation. From then on, all efforts are bent toward proving the four most fundamental theorems relating curvature and topology: the Gauss–Bonnet theorem (expressing the total curvature of a surface in term so fits topological type), the Cartan–Hadamard theorem (restricting the topology of manifolds of nonpositive curvature), Bonnet's theorem (giving analogous restrictions on manifolds of strictly positive curvature), and a special case of the Cartan–Ambrose–Hicks theorem (characterizing manifolds of constant curvature). Many other results and techniques might reasonably claim a place in an introductory Riemannian geometry course, but could not be included due to time constraints. This text is designed for a one-quarter or one-semester graduate course on Riemannian geometry. It focuses on developing an intimate acquaintance with the geometric meaning of curvature and thereby introduces and demonstrates all the main technical tools needed for a more advanced study of Riemannian manifolds. The book begins with a careful treatment of the machinery of metrics, connections, and geodesics, and then introduces the curvature tensor as a way of measuring whether a Riemannian manifold is locally equivalent to Euclidean space. Submanifold theory is developed next in order to give the curvature tensor a concrete quantitative interpretation. The remainder of the text is devoted to proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet's Theorem, and the characterization of manifolds of constant curvature. This unique volume will appeal especially to students by presenting a selective introduction to the main ideas of the subject in an easily accessible way. The material is ideal for a single course, but broad enough to provide students with a firm foundation from which to pursue research or develop applications in Riemannian geometry and other fields that use its tools.