Contents Chapter l. Affine Connections 1 1. Connection on a Manifold 1 2. Covariant Differentiation and Parallel Translation Along a Curve 3 4. Exponential Mapping and Normal Neighborhoods 7 5. Whitehead Theorem 9 Chapter 2. Covariant Differentiation. Curvature 14 1. Covariant Differentiation 14 2. The Case of Tensors of Type (r,1) 16 3. Torsion Tensor and Symmetric Connections 18 4. Geometric Meaning of the Symmetry of a Connection 20 5. Commutativity of Second Covariant Derivatives 21 6. Curvature Tensor of an Affine Connection 22 7. Space with Absolute Parallelism 24 9. Trace of the Curvature Tensor 27 10.Ricci Tensor 27 Chapter 3. Affine Mappings. Submanifolds 29 1. Affine Mappings 29 2. Affinities 32 3. Affine Coverings 33 4. Restriction of a Connection to a Submanifold 35 5. Induced Connection on a Normalized Submanifold 37 6. Gauss Formula and the Second Fundamental Form of a Normalized Submanifold 38 7. Totally Geodesic and Auto-Parallel Submanifolds 40 8. Normal Connection and the Weingarten Formula 42 9. Van der Waerden-Bortolotti Connection 42 Chapter 4. Structural Equations. Local Symmetries 44 1. Torsion and Curvature Forms 44 2. Cartan Structural Equations in Polar Coordinates 47 3. Existence of Affine Local Mappings 50 4. Locally Symmetric Affine Connection Spaces 51 5. Local Geodesic Symmetries 53 6. Semisymmetric Spaces 54 Chapter 5. Symmetric Spaces 55 1. Globally Symmetric Spaces 55 2. Germs of Smooth Mappings 55 3. Extensions of Affine Mappings 56 4. Uniqueness Theorem 58 5. Reduction of Locally Symmetric Spaces to Globally Symmetric Spaces 59 6. Properties of Symmetries in Globally Symmetric Spaces 60 7. Symmetric Spaces 62 8. Examples of Symmetric Spaces 62 9. Coincidence of Classes of Symmetric and Globally Symmetric Spaces 63 Chapter 6. Connections on Lie Groups 67 1. Invariant Construction of the Canonical Connection 67 2. Morphisms of Symmetric Spaces as Affine Mappings 69 3. Left-Invariant Connections on a Lie Group 70 4. Cartan Connections 71 5. Left Cartan Connection 73 6. Right-Invariant Vector Fields 74 7. Right Cartan Connection 76 Chapter 7. Lie Functor 77 1. Categories 77 2. Functors 78 3. Lie Functor 79 4. Kernel and Image of a Lie Group Homomorphism 80 5. Dynkin Polynomials 83 6. Dynkin Polynomials 83 7. Local Lie Groups 84 8. Bijectivity of the Lie Functor 85 Chapter 8. Affine Fields and Related Topics 87 1. Affine Fields 87 2. Dimension of the Lie Algebra of Affine Fields 89 3. Completeness of Affine Fields 91 4. Mappings of Left and Right Translation on a Symmetric Space 94 5. Derivations on Manifolds with Multiplication 95 6. Lie Algebra of Derivations 96 7. Involutive Automorphism of the Derivation Algebra of a Symmetric Spaces 97 8. Symmetric Algebras and Lie Ternaries 98 9. Lie Ternary of a Symmetric Space 100 Chapter 9. Cartan Theorem 101 1. Functor s 101 2. Comparison of the Functor s with the Lie Functor l 103 3. Properties of the Functor slo4 4. Computation of the Lie Ternary of the Space 105 5. Fundamental Group of the Quotient Space 107 6. Symmetric Space with a Given Lie Ternary 109 7. Coverings 109 8. Cartan Theorem 110 9. Identification of Homogeneous Spaces with Quotient Spaces 111 10. Translations of a Symmetric Space 112 11. Proof of the Cartan Theorem 112 Chapterlo. Palais and Kobayashi Theorems 114 1. Infinite-Dimensional Manifolds and Lie Groups 114 2. Vector Fields Induced by a Lie Group Action 114 3. Palais Theorem 117 4. Kobayashi Theorem 124 5. Affine Automorphism Group 125 6. Automorphism Group of a Symmetric Space 125 7. Translation Group of a Symmetric Space 126 Chapter 11. Lagrangians in Riemannian Spaces 127 1. Riemannian and Pseudo-Riemannian Spaces 127 2. Naturanl Parameter 142 3. Geodesics in a Riemannian Space 133 4. Simplest Problem of the Calculus of Variations 134 5. Euler-Lagrange Equations 135 6. Minimum Curves and Extremals 137 7. Regular Lagrangian 139 8. Extremals of the Energy Lagrangian 139 Chapter 12. Metric Properties of Geodesics 141 1. Length of a Curve in a Riemannian Space 141 2. Natural Parameter 142 3. Riemannian Distance and Shortest Arcs 142 4. Extremals of the Length Lagrangian 143 5. Riemannian Voordinates 144 6. Gauss Lemma 145 7. Geodesics are Locally Shortest Arcs 148 8. Smoothness of Shortest Arcs 149 9. Local Existence of Shortest Arcs 150 Chapter 13. Harmonic Functionals and Related Topics 159 1. Riemannian Volume Element 159 2. Discriminant Tensor 159 3. Foss-Weyl Formula 160 4. Vase n=2 162 5. Laplace Operator on a Riemannian Space 164 6. The Green Formulas 165 7. Existence of Harmonic Functions with a Nonzero Differential 166 8. Conjugate Harmonic Functions 170 9. Isothermal Voordinates 172 10. Semi-Cartesian Coordinates 173 11. Cartesian Coordinates 175 Chapter 14. Minimal Surfaces 176 1. Conformal Coordinates 176 3. Minimal Surfaces 178 4. Explanation of Their Name 181 5. Plateau Problem 181 6. Free Relativistic Strings 182 7. Simplest Problem of the Calculus of Variations for Functions of Two Variables 184 8. Extremals of the Area Functional 186 9. Case n=3 188 10. Representation of Minimal Surfaces Via Holomorphic Functions 189 11. Weierstrass Formulas 190 12. Adjoined Minimal Surfaces 191 Chapter 15. Curvature in Riemannian Space 193 1. Riemannian Curvature Tensor 193 2. Symmetries of the Riemannian Tensor 193 3. Riemannian Tensor as a Functional 198 4. Walker Identity and Its Consequences 199 5. Recurrent Spaces 200 6. Virtual Curvature Tensors 201 7. Reconstruction of the Bianci Tensor from Its Values on Bivectors 202 8. Sectional Curvatures 204 9. Formula for the Sectional Curvature 205 Chapter 16. Gaussian Curvature 207 1. Bianchi Tensors as Operators 207 2. Splitting of Trace-Free Tensors 208 3. Gaussian Curvature and the Scalar Curvature 209 4. Curvature Tensor for n=2 210 5. Geometric Interpretation of the Sectional Curvature 210 6. Total Curvature of a Domain on a Surface 212 7. Rotation of a Vector Field on a Curve 214 8. Rotation of the Field of Tangent Vectors 215 9. Gauss-Bonnet Theorem 218 10. Triangulated Surfaces 220 11. Gauss-Bonnet Theorem 221 Chapter 17. Some Special Tensors 223 1. Characteristic Mumbers 223 2. Euler Characteristic Number 223 3. Hodge Operator 225 4. Euler Number of a 4m-Dimensional Manifold 226 5. Euler Characteristic of a Manifold of an Arbitrary Dimension 228 6. Signature Theorem 229 7. Ricci Tensor of a Riemannian Space 230 8. Ricci Tensor of a Bianchi Tensor 231 9. Einstein and Weyl Tensors 232 10. Case n=3 234 11. Einstein Spaces 234 12. Thomas Criterion 236 Chapter 18. Surfaces with Conformal Structure 238 1. Conformal rlyansformations of a Metric 238 2. Conformal Curvature Tensor 240 3. Conformal Equivalencies 241 4. Conformally Flat Spaces 242 5. Conformally Equivalent Surfaces 243 6. Classification of Surfaces with a Conformal Structure 243 6.1 Surfaces of Parabolic Type 244 6.2 Surfaces of Elliptic Type 245 6.3 Surfaces of Hyperbolic Type 246 Chapter 19. Mappings and Submanifolds I 248 1. Locally Isometric Mapping of Riemannian Spaces 248 2. Metric Coverings 249 3. Theorem on Expanding Mappings 250 4. Isometric Mappings of Riemannian Spaces 251 5. Isometry Group of a Riemannian Space 252 6. Elliptic Geometry 252 7. Proof of Proposition 18.1 253 8. Dimension of the Isometry Group 253 9. Killing Fields 254 10. Riemannian Connection on a Submanifold of a Riemannian Space 255 11. Gauss and Weingarten Formulas for Submanifolds of Riemannian Spaces 257 12. Normal of the Mean Curvature 258 13. Gauss,Peterson-Codazzi,and Ricci Relations 259 14. Case of a Flat Ambient Space 260 Chapter 20. Submanifolds II 262 1. Locally Symmetric Submanifolds 262 2. Compact Submanifolds 267 3. Chern-Kuiper Therrem 268 4. First and Second Quadratic Forms of a Hypersurface 269 5. Hypersurfaces Whose Points are All Umbilical 271 6. Principal Curvatures of a Hypersurface 272 7. Scalar Curvature of a Hypersurface 273 8. Hypersurfaces That are Einstein Spaces 274 9. Rigidity of the Sphere 275 Chapter 21. Fundamental Forms of a Hypersurface 276 1. Sufficient Condition for Rigidity of Hypersurfaces 276 2. Hypersurfaces with a Given Second Fundamental Form 277 3. Hypersurfaces with Given First and Second Fundamental Forms 278 4. Proof of the Uniqueness 280 5. Proof of the Existence 281 6. Proof of a Local Variant of the Existence and Uniqueness Theorem 282 Chapter 22. Spaces of Constant Curvature 288 1. Spaces of Constant Curvature 288 2. Model Spaces of Constant Curvature 290 3. Model Spaces as Hypersurfaces 292 4. Isometries of Model Spaces 294 5. Fixed Points of Isometries 296 6. Riemann Theorem 296 Chapter 23. Space Forms 298 1. Space Forms 298 2. Cartan-Killing Theorem 299 3. (Pseudo-)Riemannian Symmetric Spaces 299 4. Classification of Space Forms 300 5. Spherical Forms of Even Dimension 301 6. Orientable Space Forms 302 7. Complex-Analytic and Conformal Quotient Manifolds 304 8. Riemannian Spaces with an Isometry Group of Maximal Dimension. 304 9. Their Enumeration 306 10.Complete Mobility Condition 307 Chapter 24. Four-Dimensional Manifolds 308 1. Bianchi Tensors for n=4 308 2. Matrix Representation of Bianchi Tensors for n=4 309 3. Explicit Form of Bianchi Tensors for n=4 311 4. Euler Numbers for n=4 313 5. Chern-Milnor Theorem 314 6. Sectional Curvatures of Four-Dimensional Einstein Spaces 316 7. Berger Theorem 316 8. Pontryagin Number of a Four-Dimensional Riemannian Space 317 9. Thorp Theorem 319 10. Sentenac Theorem 320 Chapter 25. Metrics on a Lie Group I 324 1. Left-Invariant Metrics on a Lie Group 324 2. Invariant Metrics on a Lie Group 324 3. Semisimple Lie Groups and Algebras 326 4. Simple Lie Groups and Algebras 329 5. Inner Derivations of Lie Algebras 329 6. Adjoint Group 331 7. Lie Groups and Algebras Without Center 332 Chapter 26. Metrics on a Lie Group II 333 1. Maurer-Cartan Forms 333 2. Left-Invariant Differential Forms 334 3. Haar Measure on a Lie group 336 4. Unimodular Lie Groups 339 5. Invariant Riemannian Metrics on a Compact Lie Group 340 6. Lie Groups with a Compact Lie Algebra 341 7. Weyl Theorem 343 Chapter 27. Jacobi Theory 344 1. Conjugate Points 344 2. Second Variation of Length 345 3. Formula for the Second Variation 346 4. Reduction of the Problem 348 5. Minimal Fields and Jacobi Fields 349 6. Jacobi Variation 351 7. Jacobi Fields and Conjugate Points 353 8. Properties of Jacobi Fields 353 9. Minimality of Normal Jacobi Fields 355 10. Proof of the Jacobi Theorem 358 Chapter 28. Some Additional Theorems I 360 1. Cut Points 360 2. Lemma on Continuity 361 3. Cut Loci and Maximal Normal Neighborhoods 362 4. Proof of Lemma 28.1 364 5. Spaces of Strictly Positive Ricci Curvature 367 6. Mayers Theorem 368 7. Spaces of Strictly Positive Sectional Curvature 369 8. Spaces of Nonpositive Sectional Curvature 370 Chapter 29. Some Additional Theorems II 371 1. Cartan Hadamard Theorem 371 2. Consequence of the Cartan Hadamard Theorem 374 3. Cartan-Killing Theorem for K=0 375 4. Bochner Theorem 375 5. Operators Ax 376 6. Infinitesimal Variant of the Bochner Theorem 378 7. Isometry Group of a Compact Space 378 Addendum 381 Chapter 30. Smooth Manifolds 381 1. Introductory Remarks 381 2. Open Sets in the Space Rn and Their Diffeomorphisms 381 3. Charts and Atlases 383 4. Maximal Atlases 385 5. Smooth Manifolds 386 6. Smooth Manifold Topology 386 7. Smooth Structures on a Topological Space 390 8. DIFF Category 391 9. Transport of Smooth Structures 392 Chapter 31. Tangent Vectors 394 1. Vectors Tangent to a Smooth Manifold 394 2. Oriented Manifolds 396 3. Differential of a Smooth Mapping 397 4. Chain Rule 398 5. Gradient of a Smooth Function 399 6. Etale Mapping Theorem 400 7. Theorem on a Local Coordinate Change 400 8. Locally Flat Mappings 401 9. Immersions and Submersions 402 Chapter 32. Submanifolds of a Smooth Manifold 404 1. Submanifolds of a Smooth Manifold 404 2. Subspace Tangent to a Submanifold 405 3. Local Representation of a Submanifold 405 4. Uniqueness of a Submanifold Structure 407 5. Case of Embedded Submanifolds 407 6. Tangent Space of a Direct Product 408 7. Theorem on the Inverse Image of a Regular Value 409 8. Solution of Sets of Equations 410 9. Embedding Theorem 411 Chapter 33. Vector and Tensor Fields. Differential Forms 413 1. Tensor Fields 413 2. Vector Fields and Derivations 416 3. Lie Algebra of Vector Fields 419 4. Integral Curves of Vector Fields 421 5. Vector Fields and Flows 422 6. Transport of Vector Fields via Diffeomorphisms 423 7. Lie Derivative of a Tensor Field 425 8. Linear Differential Forms 426 9. Differential Forms of an Arbitrary Degree 428 10. Differential Forms as Functionals on Vector Fields 429 11. Inner Product of Vector Fields and Differential Forms 430 12. Transport of a Differential Form via a Smooth Mapping 431 13. Exterior Differential 433 Chapter 34. Vector Bundles 436 1. Bundles and Their Morphisms 436 2. Vector Bundles 438 3. Sections of Vector Bundles 439 4. Morphisms of Vector Bundles 440 5. Trivial Vector Bundles 442 6. Tangent Bundles 442 7. Frame Bundles 445 8. Metricizable Bundles 447 9. ζ-Tensor Fields 447 10. Multilinear Functions andζ-Tensor Fields 449 11. Tensor Product of Vector Bundles 450 12. Generalization 450 13. Tensor Product of Sections 451 14. lnverse Image of a Vector Bundle 452 Chapter 35. Connections on Vector Bundles 454 1. Vertical Subspaces 454 2. Fields of Horizontal Subspaces 455 3. Connections and Their Forms 457 4. Inverse Image of a Connection 459 5. Horizontal Curves 460 6. Covariant Derivatives of Sections 461 7. Covariant Differentiations Along a Curve 463 8. Connections as Covariant Differentiations 463 9. Connections on Metricized Bundles 465 10. Covariant Differential 465 11. Comparison of Various Definitions of Connection 469 12. Connections on Frame Bundles 470 13. Comparison with Connections on Vector Bundles 473 Chapter 36. Curvature Tensor 475 1. Parallel Translation Along a Curve 475 2. Computation of the Parallel Translation Along a Loop 477 3. Curvature Operator at a Given Point 482 4. Translation of a Vector Along an Infinitely Small Parallelogram 484 5. Curvature Tensor 485 6. Formula for Transforming Coordinates of the Curvature Tensor 486 7. Expressing the Curvature Operator via Covariant Derivatives 487 8. Cartan Structural Equation 490 9. Bianchi Identity 491 Suggested Reading 494 Index 495
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