Lectures on the geometry of manifolds:
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
New Jersey, London u. a.
World Scientific
2007
|
| Ausgabe: | 2. ed. |
| Schlagwörter: | |
| Links: | http://www.loc.gov/catdir/toc/ecip0720/2007025469.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016276662&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Beschreibung: | Includes bibliographical references and index |
| Umfang: | XVII, 589 S. graph. Darst. |
| ISBN: | 9812708537 9789812708533 9789812778628 9812778624 |
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| 100 | 1 | |a Nicolaescu, Liviu I. |d 1964- |e Verfasser |0 (DE-588)124402887 |4 aut | |
| 245 | 1 | 0 | |a Lectures on the geometry of manifolds |c L. I. Nicolaescu |
| 246 | 1 | 3 | |a Geometry of manifolds |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New Jersey, London u. a. |b World Scientific |c 2007 | |
| 300 | |a XVII, 589 S. |b graph. Darst. | ||
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| 500 | |a Includes bibliographical references and index | ||
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| 650 | 4 | |a Manifolds (Mathematics) | |
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Datensatz im Suchindex
| _version_ | 1819320176741449728 |
|---|---|
| adam_text | LECTURES ON THE OF MANIFOLDS RY ^ * -CKV- SECOND EDITION ^ V^ R
R-RTR--F- LIVIU I NICOLAESCU UNIVERSITY OF NOTRE DAME, USA WORLD
SCIENTIFIC NEW JERSEY * LONDON * SINGAPORE * BEIJING * SHANGHAI * HONG
KONG * TAIPEI *. CHENNAI CONTENTS PREFACE VII 1. MANIFOLDS 1 1.1
PRELIMINARIES 1 1.1.1 SPACE AND COORDINATIZATION 1 1.1.2 THE IMPLICIT
FUNCTION THEOREM 3 1.2 SMOOTH MANIFOLDS 6 1.2.1 BASIC DEFINITIONS 6
1.2.2 PARTITIONS OF UNITY 9 1.2.3 EXAMPLES 9 1.2.4 HOW MANY MANIFOLDS
ARE THERE? 19 2. NATURAL CONSTRUCTIONS ON MANIFOLDS 23 2.1 THE TANGENT
BUNDLE 23 2.1.1 TANGENT SPACES 23 2.1.2 THE TANGENT BUNDLE 26 2.1.3
SARD S THEOREM 28 2.1.4 VECTOR BUNDLES 32 2.1.5 SOME EXAMPLES OF VECTOR
BUNDLES 37 1 2.2 A LINEAR ALGEBRA INTERLUDE 41 2.2.1 TENSOR PRODUCTS 41
2.2.2 SYMMETRIC AND SKEW-SYMMETRIC TENSORS 46 2.2.3 THE SUPER SLANG 53
2.2.4 DUALITY 56 2.2.5 SOME COMPLEX LINEAR ALGEBRA 64 2.3 TENSOR FIELDS
69 2.3.1 OPERATIONS WITH VECTOR BUNDLES 69 2.3.2 TENSOR FIELDS 70 2.3.3
FIBER BUNDLES 74 XIV LECTURES ON THE GEOMETRY OF MANIFOLDS 3. CALCULUS
ON MANIFOLDS 81 3.1 THE LIE DERIVATIVE 81 3.1.1 FLOWS ON MANIFOLDS 81
3.1.2 THE LIE DERIVATIVE 83 3.1.3 EXAMPLES 88 3.2 DERIVATIONS OF FI #
(M) 91 3.2.1 THE EXTERIOR DERIVATIVE 91 3.2.2 EXAMPLES 96 3.3
CONNECTIONS ON VECTOR BUNDLES 97 3.3.1 COVARIANT DERIVATIVES 97 3.3.2
PARALLEL TRANSPORT 102 3.3.3 THE CURVATURE OF A CONNECTION 104 3.3.4
HOLONOMY 106 3.3.5 THE BIANCHI IDENTITIES . . . 110 3.3.6 CONNECTIONS ON
TANGENT BUNDLES ILL 3.4 INTEGRATION ON MANIFOLDS 113 3.4.1 INTEGRATION
OF 1-DENSITIES 113 3.4.2 ORIENTABILITY AND INTEGRATION OF DIFFERENTIAL
FORMS 118 3.4.3 STOKES FORMULA 126 3.4.4 REPRESENTATIONS AND CHARACTERS
OF COMPACT LIE GROUPS . . . 130 3.4.5 FIBERED CALCULUS 137 4. RIEMANNIAN
GEOMETRY 141 4.1 METRIC PROPERTIES 141 4.1.1 DEFINITIONS AND EXAMPLES
141 4.1.2 THE LEVI-CIVITA CONNECTION * * * 145 4.1.3 THE EXPONENTIAL MAP
AND NORMAL COORDINATES 150 4.1.4 THE LENGTH MINIMIZING PROPERTY OF
GEODESIES 152 4.1.5 CALCULUS ON RIEMANN MANIFOLDS 158 4.2 THE RIEMANN
CURVATURE 168 4.2.1 DEFINITIONS AND PROPERTIES 168 4.2.2 EXAMPLES 172
4.2.3 CARTAN S MOVING FRAME METHOD 174 4.2.4 THE GEOMETRY OF
SUBMANIFOLDS 178 4.2.5 THE GAUSS-BONNET THEOREM FOR ORIENTED SURFACES
184 5. ELEMENTS OF THE CALCULUS OF VARIATIONS 193 5.1 THE LEAST ACTION
PRINCIPLE 193 5.1.1 THE 1-DIMENSIONAL EULER-LAGRANGE EQUATIONS . . . . .
. . . 193 5.1.2 NOETHER S CONSERVATION PRINCIPLE 199 5.2 THE VARIATIONAL
THEORY OF GEODESIES 203 5.2.1 VARIATIONAL FORMULAE 203 CONTENTS XV 5.2.2
JACOBI FIELDS 207 6. THE FUNDAMENTAL GROUP AND COVERING SPACES 215 6.1
THE FUNDAMENTAL GROUP 216 6.1.1 BASIC NOTIONS 216 6.1.2 OF CATEGORIES
AND FUNCTORS 220 6.2 COVERING SPACES 222 6.2.1 DEFINITIONS AND EXAMPLES
222 6.2.2 UNIQUE LIFTING PROPERTY 224 6.2.3 HOMOTOPY LIFTING PROPERTY
225 6.2.4 ON THE EXISTENCE OF LIFTS 226 6.2.5 THE UNIVERSAL COVER AND
THE FUNDAMENTAL GROUP 228 7. COHOMOLOGY 231 7.1 DERHAM COHOMOLOGY 231
7.1.1 SPECULATIONS AROUND THE POINCARE LEMMA 231 7.1.2 CECH VS. DERHAM
235 7.1.3 VERY LITTLE HOMOLOGICAL ALGEBRA 237 7.1.4 FUNCTORIAL
PROPERTIES OF THE DERHAM COHOMOLOGY 244 7.1.5 SOME SIMPLE EXAMPLES 247
7.1.6 THE MAYER-VIETORIS PRINCIPLE 249 7.1.7 THE KIINNETH FORMULA 253
7.2 THE POINCARE DUALITY 255 7.2.1 COHOMOLOGY WITH COMPACT SUPPORTS 255
7.2.2 THE POINCARE DUALITY 259 7.3 INTERSECTION THEORY 263 7.3.1 CYCLES
AND THEIR DUALS 263 7.3.2 INTERSECTION THEORY 268 7.3.3 THE TOPOLOGICAL
DEGREE 274 7.3.4 THOM ISOMORPHISM THEOREM 276 7.3.5 GAUSS-BONNET
REVISITED 279 7.4 SYMMETRY AND TOPOLOGY 283 7.4.1 SYMMETRIC SPACES 284
7.4.2 SYMMETRY AND COHOMOLOGY 287 7.4.3 THE COHOMOLOGY OF COMPACT LIE
GROUPS 290 7.4.4 INVARIANT FORMS ON GRASSMANNIANS AND WEYL S INTEGRAL
FORMULA 292 7.4.5 THE POINCARE POLYNOMIAL OF A COMPLEX GRASSMANNIAN . .
. 299 7.5 CECH COHOMOLOGY 305 7.5.1 SHEAVES AND PRESHEAVES 305 7.5.2
CECH COHOMOLOGY 310 XVI LECTURES ON THE GEOMETRY OF MANIFOLDS 8.
CHARACTERISTIC CLASSES 321 8.1 CHERN-WEIL THEORY 321 8.1.1 CONNECTIONS
ON PRINCIPAL G : BUNDLES 321 8.1.2 G-VECTOR BUNDLES 327 8.1.3 INVARIANT
POLYNOMIALS 328 8.1.4 THE CHERN-WEIL THEORY 331 8.2 IMPORTANT EXAMPLES
335 8.2.1 THE INVARIANTS OF THE TORUS T N 335 8.2.2 CHERN CLASSES
.... ..- 335 8.2.3 PONTRYAGIN CLASSES ... 338 8.2.4 THE EULER CLASS ,
340 8.2.5 UNIVERSAL CLASSES 344 8.3 COMPUTING CHARACTERISTIC CLASSES 350
8.3.1 REDUCTIONS 351 8.3.2 THE GAUSS-BONNET-CHERN THEOREM 356 9.
CLASSICAL INTEGRAL GEOMETRY 367 9.1 THE INTEGRAL GEOMETRY OF REAL
GRASSMANNIANS 367 9.1.1 CO-AREA FORMULAE 367 9.1.2 INVARIANT MEASURES ON
LINEAR GRASSMANNIANS 378 9.1.3 AFFINE GRASSMANNIANS 388 9.2 GAUSS-BONNET
AGAIN?!? 390 9.2.1 THE SHAPE OPERATOR AND THE SECOND FUNDAMENTAL FORM OF
A SUBMANIFOLD IN R 391 9.2.2 THE GAUSS-BONNET THEOREM FOR HYPERSURFACES
OF A EUCLIDEAN SPACE 393 9.2.3 GAUSS-BONNET THEOREM FOR DOMAINS OF A
EUCLIDEAN SPACE . 399 9.3 CURVATURE MEASURES 402 9.3.1 TAME GEOMETRY 402
9.3.2 INVARIANTS OF THE ORTHOGONAL GROUP 408 9.3.3 THE TUBE FORMULA AND
CURVATURE MEASURES 412 9.3.4 TUBE FORMULA =$* GAUSS-BONNET FORMULA FOR
ARBITRARY SUBMANIFOLDS 423 9.3.5 CURVATURE MEASURES OF DOMAINS OF A
EUCLIDEAN SPACE . . . 425 9.3.6 CROFTON FORMULAE FOR DOMAINS OF A
EUCLIDEAN SPACE . . . . . 427 9.3.7 CROFTON FORMULAE FOR SUBMANIFOLDS OF
A EUCLIDEAN SPACE . . 438 10. ELLIPTIC EQUATIONS ON MANIFOLDS 445 10.1
PARTIAL DIFFERENTIAL OPERATORS: ALGEBRAIC ASPECTS 445 10.1.1 BASIC
NOTIONS . . 445 10.1.2 EXAMPLES 451 10.1.3 FORMAL ADJOINTS 454 CONTENTS
XVII 10.2 FUNCTIONAL FRAMEWORK 460 10.2.1 SOBOLEV SPACES IN R N 460
10.2.2 EMBEDDING THEOREMS: INTEGRABILITY PROPERTIES 467 10.2.3 EMBEDDING
THEOREMS: DIFFERENTIABILITY PROPERTIES 471 10.2.4 FUNCTIONAL SPACES ON
MANIFOLDS 475 10.3 ELLIPTIC PARTIAL DIFFERENTIAL OPERATORS: ANALYTIC
ASPECTS 479 10.3.1 ELLIPTIC ESTIMATES IN R N 480 10.3.2 ELLIPTIC
REGULARITY 485 10.3.3 AN APPLICATION: PRESCRIBING THE CURVATURE OF
SURFACES . . . 490 10.4 ELLIPTIC OPERATORS ON COMPACT MANIFOLDS 500
10.4.1 FREDHOLM THEORY 500 10.4.2 SPECTRAL THEORY 510 10.4.3 HODGE
THEORY 514 11. DIRAC OPERATORS 519 11.1 THE STRUCTURE OF DIRAC OPERATORS
519 11.1.1 BASIC DEFINITIONS AND EXAMPLES 519 11.1.2 CLIFFORD ALGEBRAS
522 11.1.3 CLIFFORD MODULES: THE EVEN CASE 526 11.1.4 CLIFFORD MODULES:
THE ODD CASE 530 11.1.5 A LOOK AHEAD 531 11.1.6 SPIN 533 11.1.7 SPIN C
542 11.1.8 LOW DIMENSIONAL EXAMPLES 544 11.1.9 DIRAC BUNDLES 549 11.2
FUNDAMENTAL EXAMPLES 553 11.2.1 THE HODGE-DERHAM OPERATOR 553 11.2.2 THE
HODGE-DOLBEAULT OPERATOR 558 11.2.3 THE SPIN DIRAC OPERATOR 564 11.2.4
THE SPIN 0 DIRAC OPERATOR 570 BIBLIOGRAPHY 579 INDEX 583
|
| any_adam_object | 1 |
| author | Nicolaescu, Liviu I. 1964- |
| author_GND | (DE-588)124402887 |
| author_facet | Nicolaescu, Liviu I. 1964- |
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| ctrlnum | (OCoLC)144773920 (DE-599)DNB 2007025469 |
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| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/62 |
| dewey-search | 516.3/62 |
| dewey-sort | 3516.3 262 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV023073540 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T13:08:01Z |
| institution | BVB |
| isbn | 9812708537 9789812708533 9789812778628 9812778624 |
| language | English |
| lccn | 2007025469 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016276662 |
| oclc_num | 144773920 |
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| owner | DE-384 DE-634 DE-11 DE-20 |
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| physical | XVII, 589 S. graph. Darst. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | World Scientific |
| record_format | marc |
| spellingShingle | Nicolaescu, Liviu I. 1964- Lectures on the geometry of manifolds Geometry, Differential Manifolds (Mathematics) Differentialgeometrie (DE-588)4012248-7 gnd Mannigfaltigkeit (DE-588)4037379-4 gnd Glatte Mannigfaltigkeit (DE-588)4157471-0 gnd |
| subject_GND | (DE-588)4012248-7 (DE-588)4037379-4 (DE-588)4157471-0 |
| title | Lectures on the geometry of manifolds |
| title_alt | Geometry of manifolds |
| title_auth | Lectures on the geometry of manifolds |
| title_exact_search | Lectures on the geometry of manifolds |
| title_full | Lectures on the geometry of manifolds L. I. Nicolaescu |
| title_fullStr | Lectures on the geometry of manifolds L. I. Nicolaescu |
| title_full_unstemmed | Lectures on the geometry of manifolds L. I. Nicolaescu |
| title_short | Lectures on the geometry of manifolds |
| title_sort | lectures on the geometry of manifolds |
| topic | Geometry, Differential Manifolds (Mathematics) Differentialgeometrie (DE-588)4012248-7 gnd Mannigfaltigkeit (DE-588)4037379-4 gnd Glatte Mannigfaltigkeit (DE-588)4157471-0 gnd |
| topic_facet | Geometry, Differential Manifolds (Mathematics) Differentialgeometrie Mannigfaltigkeit Glatte Mannigfaltigkeit |
| url | http://www.loc.gov/catdir/toc/ecip0720/2007025469.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016276662&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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