Verfügbarkeit: Fundamentals of differential geometry

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Titel:	Fundamentals of differential geometry
Von:	Serge Lang
Person:	Lang, Serge 1927-2005 Verfasser aut
Hauptverfasser:	Lang, Serge 1927-2005 (VerfasserIn)
Format:	Buch
Sprache:	Englisch
Veröffentlicht:	New York [u.a.] Springer 1999
Schriftenreihe:	Graduate texts in mathematics 191
Schlagworte:	Differentialgeometrie Differentialtopologie Differenzierbare Mannigfaltigkeit Geometry, Differential Spektraltheorie Einführung
Online-Zugang:	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008418595&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Beschreibung:	Literaturverz. S. 523 - 530
Beschreibung:	XVII, 535 S. graph. Darst.
ISBN:	038798593X 9780387985930

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adam_text	Contents Foreword.................................................................................................................... v Acknowledgments.................................................................................................... xi PARTI Genera! Differential Theory............................................................................. 1 CHAPTER I Differential Calculus................................................................................................ §1. §2. §3. §4. §5. Categories.......................................................................................................... Topological Vector Spaces............................................................................. Derivatives and Composition of Maps.......................................................... Integration and Taylor’s Formula................................................................. The Inverse Mapping Theorem...................................................................... 3 4 5 8 12 15 CHAPTER II Manifolds.................................................................................................................. 22 §1. §2. §3. §4. 22 25 33 39 Atlases, Charts, Morphisms........................................................................... Submanifolds, Immersions, Submersions...................................................... Partitions of Unity............................................................................................ Manifolds with Boundary............................................................................... CHAPTER III Vector Bundles......................................................................................................... 43 §1. Definition, Pull Backs...................................................................................... §2. The Tangent Bundle........................................................................................ §3. Exact Sequences of Bundles............................................................................ 43 51 52 xiii CONTENTS XIV §4. Operations on Vector Bundles......................................................................... §5. Splitting of Vector Bundles.............................................................................. 58 63 CHAPTER IV Vector Fields and Differential Equations................................................................ §1. §2. §3. §4. §5. §6. Existence Theorem for Differential Equations.............................................. Vector Fields, Curves, and Flows................................................................... Sprays............................................................................................................... The Flow of a Spray and the Exponential Map............................................ Existence of Tubular Neighborhoods............................................................ Uniqueness of Tubular Neighborhoods......................................................... 66 67 88 96 105 110 112 CHAPTER V Operations on Vector Fields and Differential Forms............................................ 116 §1. §2. §3. §4. §5. §6. §7. §8. 116 122 124 137 139 143 149 151 Vector Fields, Differential Operators, Brackets............................................ Lie Derivative................................................................................................... Exterior Derivative.......................................................................................... The Poincaré Lemma....................................................................................... Contractions and Lie Derivative..................................................................... Vector Fields and 1-Forms Under Self Duality............................................ The Canonical 2-Form..................................................................................... Darboux’s Theorem......................................................................................... CHAPTER VI The Theorem of Frobenius..................................................................................... §1. Statement of the Theorem................................................................................ §2. Differential Equations Depending on a Parameter....................................... §3. Proof of the Theorem....................................................................................... §4. The Global Formulation.................................................................................. §5. Lie Groups and Subgroups.............................................................................. 155 155 160 161 162 165 PART II Metrics, Covariant Derivatives, and Riemannian Geometry..................... 171 CHAPTER VII Metrics...................................................................................................................... 173 §1. §2. §3. §4. §5. §6. §7. 173 177 180 184 186 189 192 Definition and Functoriality............................................................................ The Hilbert Group............................................................................................ Reduction to the Hilbert Group..................................................................... Hilbertian Tubular Neighborhoods................................................................ The Morse-Palais Lemma.............................................................................. The Riemannian Distance................................................................................ The Canonical Spray....................................................................................... CHAPTER VIII Covariant Derivatives and Geodesics..................................................................... §1. Basic Properties................................................................................................ 196 196 CONTENTS §2. §3. §4. §5. §6. Sprays and Covariant Derivatives.................................................................. Derivative Along a Curve and Parallelism..................................................... The Metric Derivative..................................................................................... More Local Results on the Exponential Map................................................ Riemannian Geodesic Length and Completeness......................................... XV 199 204 209 215 221 CHAPTER IX Curvature.................................................................................................................. 231 §1. §2. §3. §4. §5. 231 239 246 255 263 The Riemann Tensor....................................................................................... Jacobi Lifts......................................................................................................... Application of Jacobi Lifts to Texpr.............................................................. Convexity Theorems......................................................................................... Taylor Expansions............................................................................................. CHAPTER X Jacobi Lilts and Tensorial Splitting of the Double Tangent Bundle..................... 267 §1. §2. §3. §4. §5. §6. 267 271 276 279 286 291 Convexity of Jacobi Lifts.................................................................................. Global Tubular Neighborhood of a Totally Geodesic Submanifold........... More Convexity and Comparison Results..................................................... Splitting of the Double Tangent Bundle......................................................... Tensorial Derivative of a Curve in TX and of the Exponential Map.......... The Flow and the Tensorial Derivative......................................................... CHAPTER XI Curvature and the Variation Formula..................................................................... 294 §1. §2. §3. §4. §5. 294 304 309 315 318 The Index Form, Variations, and the Second Variation Formula.............. Growth of a Jacobi Lift.................................................................................... The Semi Parallelogram Law and Negative Curvature...................... Totally Geodesic Submanifolds....................................................................... Rauch Comparison Theorem........................................................................... CHAPTER XII An Example of Seminegative Curvature................................................................ 322 §1. Pos„(R) as a Riemannian Manifold................................................................ §2. The Metric Increasing Property of the Exponential Map............................ §3. Totally Geodesic and Symmetric Submanifolds............................................ 322 327 332 CHAPTER XIII Automorphisms and Symmetries........................................................................... 339 §1. §2. §3. §4. §5. §6. 342 347 351 354 358 365 The Tensorial Second Derivative.................................................................... Alternative Definitions of Killing Fields......................................................... Metric Killing Fields......................................................................................... Lie Algebra Properties of Killing Fields......................................................... Symmetric Spaces.............................................................................................. Parallelism and the Riemann Tensor.............................................................. CONTENTS XVI CHAPTER XIV Immersions and Submersions................................................................................ 369 §1. §2. §3. §4. §5. §6. 369 376 383 387 390 393 The Covariant Derivative on a Submanifold................................................ The Hessian and Laplacian on a Submanifold.........;................................... The Covariant Derivative on a Riemannian Submersion........................... The Hessian and Laplacian on a Riemannian Submersion........................ The Riemann Tensor on Submanifolds......................................................... The Riemann Tensor on a Riemannian Submersion.................................. PART III Volume Forms and Integration......................................................................... 395 CHAPTER XV Volume Forms........................................................................................................... 397 §1. §2. §3. §4. §5. §6. §7. §8. 397 407 412 418 424 428 435 440 Volume Forms and the Divergence................................................................ Covariant Derivatives....................................................................................... The Jacobian Determinant of the Exponential Map.................................... The Hodge Star on Forms.............................................................................. Hodge Decomposition of Differential Forms................................................ Volume Forms in a Submersion..................................................................... Volume Forms on Lie Groups and Homogeneous Spaces........................... Homogeneously Fibered Submersions............................................................. CHAPTER XVI Integration of Differential Forms............................................................................. 448 §1. §2. §3. §4. §5. 448 453 461 463 471 Sets of Measure 0.............................................................................................. Change of Variables Formula......................................................................... Orientation......................................................................................................... The Measure Associated with a Differential Form....................................... Homogeneous Spaces....................................................................................... CHAPTER XVII Stokes’ Theorem....................................................................................................... 475 §1. Stokes’ Theorem for a Rectangular Simplex.................................................. §2. Stokes’ Theorem on a Manifold..................................................................... §3. Stokes’ Theorem with Singularities................................................................ 475 478 482 CHAPTER XVIII Applications of Stokes’ Theorem............................................................................ 489 §1. §2. §3. §4. §5. §6. 489 496 497 501 503 507 The Maximal de Rham Cohomology........................................................... Moser’s Theorem............................................................................................. The Divergence Theorem............................................................................... The Adjoint of d for Higher Degree Forms................................................... Cauchy’s Theorem........................................................................................... The Residue Theorem...................................................................................... CONTENTS XVÍÍ APPENDIX The Spectral Theorem............................................................................................... 511 §1. Hilbert Space................................................................................................. §2. Functionals and Operators........................................................................... §3. Hermitian Operators..................................................................................... 511 512 515 Bibliography............................................................................................................. 523 Index......................................................................................................................... 531
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spellingShingle	Lang, Serge 1927-2005 Fundamentals of differential geometry Graduate texts in mathematics Differentialgeometrie Differentialtopologie Differenzierbare Mannigfaltigkeit Geometry, Differential Differentialtopologie (DE-588)4012255-4 gnd Differenzierbare Mannigfaltigkeit (DE-588)4012269-4 gnd Spektraltheorie (DE-588)4116561-5 gnd Differentialgeometrie (DE-588)4012248-7 gnd
subject_GND	(DE-588)4012255-4 (DE-588)4012269-4 (DE-588)4116561-5 (DE-588)4012248-7 (DE-588)4151278-9
title	Fundamentals of differential geometry
title_auth	Fundamentals of differential geometry
title_exact_search	Fundamentals of differential geometry
title_full	Fundamentals of differential geometry Serge Lang
title_fullStr	Fundamentals of differential geometry Serge Lang
title_full_unstemmed	Fundamentals of differential geometry Serge Lang
title_short	Fundamentals of differential geometry
title_sort	fundamentals of differential geometry
topic	Differentialgeometrie Differentialtopologie Differenzierbare Mannigfaltigkeit Geometry, Differential Differentialtopologie (DE-588)4012255-4 gnd Differenzierbare Mannigfaltigkeit (DE-588)4012269-4 gnd Spektraltheorie (DE-588)4116561-5 gnd Differentialgeometrie (DE-588)4012248-7 gnd
topic_facet	Differentialgeometrie Differentialtopologie Differenzierbare Mannigfaltigkeit Geometry, Differential Spektraltheorie Einführung
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volume_link	(DE-604)BV000000067
work_keys_str_mv	AT langserge fundamentalsofdifferentialgeometry

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