I. Introduction to Manifolds l.Preliminary Comments on Rn 1 2.Rn and Euclidean Space 4 3.Topological Manifolds 6 4.Further Examples of Manifolds. Cutting and Pasting 11 5.Abstract Manifolds. Some Examples 14 II. Functions of Several Variables and Mappings 1.Differentiability for Functions of Several Variables 20 2.Differentiability of Mappings and Jacobians 25 3.The Space of Tangent Vectors at a Point of Rn 29 4.Another Definition of Ta(Rn) 32 5.Vector Fields on Open Subsets of Rn 36 6.The Inverse Function Theorem 41 7.The Rank of a Mapping 46 III. Differentiable Manifolds and Submanifolds 1.The Definition of a Differentiable Manifold 52 2.Further Examples 59 3.Differentiable Functions and Mappings 65 4.Rank of a Mapping, Immersions 68 5.Submanifolds 74 6.Lie Groups 80 7.The Action of a Lie Group on a Manifold. Transformation Groups 87 8.The Action of a Discrete Group on a Manifold 93 9.Covering Manifolds 98 IV. Vector Fields on a Manifold 1.The Tangent Space at a Point of a Manifold 104 2.Vector Fields 113 3.One-Parameter and Local One-Parameter Groups Acting on a Manifold 119 4.The Existence Theorem for Ordinary Differential Equations 127 5.Some Examples of One-Parameter Groups Acting on a Manifold 135 6.One-Parameter Subgroups of Lie Groups 142 7.The Lie Algebra of Vector Fields on a Manifold 146 8.Frobenius's Theorem 153 9.Homogeneous Spaces 160 V. Tensors and Tensor Fields on Manifolds 1.Tangent Covectors 171 Covectors on Manifolds 172 Covector Fields and Mappings 174 2.Bilinear Forms. The Riemannian Metric 177 3.Riemannian Manifolds as Metric Spaces 181 4.Partitions of Unity 186 Some Applications of the Partition of Unity 188 5.Tensor Fields 192 Tensors on a Vector Space 192 Tensor Fields 194 Mappings and Covariant Tensors 195 The Symmetrizing and Alternating Transformations 196 6.Multiplication of Tensors 199 Multiplication of Tensors on a Vector Space 199 Multiplication of Tensor Fields 201 Exterior Multiplication of Alternating Tensors 202 The Exterior Algebra on Manifolds 206 7.Orientation of Manifolds and the Volume Element 207 8.Exterior Differentiation 212 An Application to Frobenius's Theorem 216 VI. Integration on Manifolds 1.Integration in R" Domains of Integration 223 Basic Properties of the Riemann Integral 224 2.A Generalization to Manifolds 229 Integration on Riemannian Manifolds 232 3.Integration on Lie Groups 237 4.Manifolds with Boundary 243 5.Stokes's Theorem for Manifolds 251 6.Homotopy of Mappings. The Fundamental Group 258 Homotopy of Paths and Loops. The Fundamental Group 259 7.Some Applications of Differential Forms. The de Rham Groups 265 The Homotopy Operator 268 8.Some Further Applications of de Rham Groups 272 The de Rham Groups of Lie Groups 276 9.Covering Spaces and Fundamental Group 280 VII. Differentiation on Riemannian Manifolds l.Differentiation of Vector Fields along Curves in Rn 289 The Geometry of Space Curves 292 Curvature of Plane Curves 296 2.Differentiation of Vector Fields on Submanifolds of Rn 298 Formulas for Covariant Derivatives 303 ▽xpY and Differentiation of Vector Fields 305 3.Differentiation on Riemannian Manifolds 308 Constant Vector Fields and Parallel Displacement 314 4.Addenda to the Theory of Differentiation on a Manifold 316 The Curvature Tensor 316 The Riemannian Connection and Exterior Differential Forms 319 5.Geodesic Curves on Riemannian Manifolds 321 6.The Tangent Bundle and Exponential Mapping. Normal Coordinates 326 7.Some Further Properties of Geodesics 332 8.Symmetric Riemannian Manifolds 340 9.Some Examples 346 VIII. Curvature 1.The Geometry of Surfaces in E3 355 The Principal Curvatures at a Point of a Surface 359 2.The Gaussian and Mean Curvatures of a Surface 363 The Theorema Egregium of Gauss 366 3.Basic Properties of the Riemann Curvature Tensor 371 4.Curvature Forms and the Equations of Structure 378 5.Differentiation of Covariant Tensor Fields 384 6.Manifolds of Constant Curvature 391 Spaces of Positive Curvature 394 Spaces of Zero Curvature 396 Spaces of Constant Negative Curvature 397 REFERENCES 403 INDEX 41 1
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