An introduction to Riemann-Finsler geometry:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2000
|
| Series: | Graduate texts in mathematics
200 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturangaben |
| Physical Description: | XX, 431 S. graph. Darst. : 24 cm |
| ISBN: | 038798948X |
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| _version_ | 1804128010588127232 |
|---|---|
| adam_text | D. BAO S.-S. CHERN Z. SHEN AN INTRODUCTION TO RIEMANN-FINSLER GEOMETRY
WITH 20 ILLUSTRATIONS SPRINGER CONTENTS PREFACE ACKNOWLEDGMENTS VN XIII
PART ONE FINSLER MANIFOLDS AND THEIR CURVATURE CHAPTER 1 FINSLER
MANIFOLDS AND THE FUNDAMENTALS OF MINKOWSKI NORMS 1 1.0 PHYSICAL
MOTIVATIONS 1 1.1 FINSLER STRUCTURES: DEFINITIONS AND CONVENTIONS 2 1.2
TWO BASIC PROPERTIES OF MINKOWSKI NORMS 5 1.2 A. EULER S THEOREM 5 1.2
B. A FUNDAMENTAL INEQUALITY 6 1.2 C. INTERPRETATIONS OF THE FUNDAMENTAL
INEQUALITY 9 1.3 EXPLICIT EXAMPLES OF FINSLER MANIFOLDS 14 1.3 A.
MINKOWSKI AND LOCALLY MINKOWSKI SPACES 14 1.3 B. RIEMANNIAN MANIFOLDS 15
1.3 C. RANDERS SPACES 17 1.3 D. BERWALD SPACES 18 1.3 E. FINSLER SPACES
OF CONSTANT FLAG CURVATURE 20 1.4 THE FUNDAMENTAL TENSOR AND THE CARTAN
TENSOR 22 * REFERENCES FOR CHAPTER 1 25 CHAPTER 2 THE CHERN CONNECTION
27 2.0 PROLOGUE 27 2.1 THE VECTOR BUNDLE TT*TM AND RELATED OBJECTS 28
2.2 COORDINATE BASES VERSUS SPECIAL ORTHONORMAL BASES 31 2.3 THE
NONLINEAR CONNECTION ON THE MANIFOLD TM 0 33 2.4 THE CHERN CONNECTION
ON -K*TM 37 XVI CONTENTS 2.5 INDEX GYMNASTICS 44 2.5 A. THE SLASH (...)|
S AND THE SEMICOLON (...) ;S 44 2.5 B. COVARIANT DERIVATIVES OF THE
FUNDAMENTAL TENSOR G 45 2.5 C. COVARIANT DERIVATIVES OF THE
DISTINGUISHED 46 * REFERENCES FOR CHAPTER 2 48 CHAPTER 3 CURVATURE AND
SCHUR S LEMMA 49 3.1 CONVENTIONS AND THE HH-, HV-, UU-CURVATURES 49 3.2
FIRST BIANCHI IDENTITIES FROM TORSION FREENESS 50 3.3 FORMULAS FOR R AND
P IN NATURAL COORDINATES 52 3.4 FIRST BIANCHI IDENTITIES FROM ALMOST
^-COMPATIBILITY 54 3.4 A. CONSEQUENCES FROM THE DX K A DX L TERMS 55 3.4
B. CONSEQUENCES FROM THE DX K A J?6Y L TERMS 55 3.4 C. CONSEQUENCES FROM
THE J^SY K A JSY 1 TERMS 56 3.5 SECOND BIANCHI IDENTITIES 58 3.6
INTERCHANGE FORMULAS OR RICCI IDENTITIES 61 3.7 LIE BRACKETS AMONG THE ~
AND THE F-§^ 62 3.8 DERIVATIVES OF THE GEODESIC SPRAY COEFFICIENTS G X
65 3.9 THE FLAG CURVATURE 67 3.9 A. ITS DEFINITION AND ITS PREDECESSOR
68 3.9 B. AN INTERESTING FAMILY OF EXAMPLES OF NUMATA TYPE 70 3.10
SCHUR S LEMMA 75 * REFERENCES FOR CHAPTER 3 80 CHAPTER 4 FINSLER
SURFACES AND A GENERALIZED GAUSS*BONNET THEOREM 81 4.0 PROLOGUE 81 4.1
MINKOWSKI PLANES AND A USEFUL BASIS 82 4.1 A. RUND S DIFFERENTIAL
EQUATION AND ITS CONSEQUENCE 83 4.1 B. A CRITERION FOR CHECKING STRONG
CONVEXITY 86 4.2 THE EQUIVALENCE PROBLEM FOR MINKOWSKI PLANES 90 4.3 THE
BERWALD FRAME AND OUR GEOMETRICAL SETUP ON SM 92 4.4 THE CHERN
CONNECTION AND THE INVARIANTS I, J, K 95 4.5 THE RIEMANNIAN ARC LENGTH
OF THE INDICATRIX 101 4.6 A GAUSS-BONNET THEOREM FOR LANDSBERG SURFACES
105 * REFERENCES FOR CHAPTER 4 110 CONTENTS PART TWO CALCULUS OF
VARIATIONS AND COMPARISON THEOREMS 111 CHAPTER 5 VARIATIONS OF ARC
LENGTH, JACOBI FIELDS, THE EFFECT OF CURVATURE ILL 5.1 THE FIRST
VARIATION OF ARC LENGTH ILL 5.2 THE SECOND VARIATION OF ARC LENGTH 119
5.3 GEODESIES AND THE EXPONENTIAL MAP 125 5.4 JACOBI FIELDS 129 5.5 HOW
THE FLAG CURVATURE S SIGN INFLUENCES GEODESIC RAYS 135 * REFERENCES FOR
CHAPTER 5 138 CHAPTER 6 THE GAUSS LEMMA AND THE HOPF-RINOW THEOREM 139
6.1 THE GAUSS LEMMA 139 6.1 A. THE GAUSS LEMMA PROPER 140 6.1 B. AN
ALTERNATIVE FORM OF THE LEMMA 142 6.1 C. IS THE EXPONENTIAL MAP EVER A
LOCAL ISOMETRY? 143 6.2 FINSLER MANIFOLDS AND METRIC SPACES 145 6.2 A. A
USEFUL TECHNICAL LEMMA 146 6.2 B. FORWARD METRIC BALLS AND METRIC
SPHERES 148 6.2 C. THE MANIFOLD TOPOLOGY VERSUS THE METRIC TOPOLOGY ...
149 6.2 D. FORWARD CAUCHY SEQUENCES, FORWARD COMPLETENESS ... 151 6.3
SHORT GEODESIES ARE MINIMIZING 155 6.4 THE SMOOTHNESS OF DISTANCE
FUNCTIONS 161 6.4 A. ON MINKOWSKI SPACES 161 6.4 B. ON FINSLER MANIFOLDS
162 6.5 LONG MINIMIZING GEODESIES 164 6.6 THE HOPF-RINOW THEOREM 168 *
REFERENCES FOR CHAPTER 6 172 CHAPTER 7 THE INDEX FORM AND THE
BONNET-MYERS THEOREM 173 7.1 CONJUGATE POINTS 173 7.2 THE INDEX FORM 176
7.3 WHAT HAPPENS IN THE ABSENCE OF CONJUGATE POINTS? 179 7.3 A.
GEODESIES ARE SHORTEST AMONG NEARBY CURVES 179 7.3 B. A BASIC INDEX
LEMMA 182 7.4 WHAT HAPPENS IF CONJUGATE POINTS ARE PRESENT? 184 7.5 THE
CUT POINT VERSUS THE FIRST CONJUGATE POINT 186 XVIII CONTENTS 7.6 RICCI
CURVATURES 190 7.6 A. THE RICCI SCALAR RIC AND THE RICCI TENSOR RICIJ
191 7.6 B. THE INTERPLAY BETWEEN RIC AND RIC TJ 192 7.7 THE BONNET-MYERS
THEOREM 194 * REFERENCES FOR CHAPTER 7 198 CHAPTER 8 THE CUT AND
CONJUGATE LOCI, AND SYNGE S THEOREM 199 8.1 DEFINITIONS 199 8.2 THE CUT
POINT AND THE FIRST CONJUGATE POINT 201 8.3 SOME CONSEQUENCES OF THE
INVERSE FUNCTION THEOREM 204 8.4 THE MANNER IN WHICH CY AND I Y DEPEND
ON Y 206 8.5 GENERIC PROPERTIES OF THE CUT LOCUS CUT X 208 8.6
ADDITIONAL PROPERTIES OF CUT X WHEN M IS COMPACT 211 8.7 SHORTEST
GEODESIES WITHIN HOMOTOPY CLASSES 213 8.8 SYNGE S THEOREM 221 *
REFERENCES FOR CHAPTER 8 224 CHAPTER 9 THE CARTAN*HADAMARD THEOREM AND
RAUCH S FIRST THEOREM 225 9.1 ESTIMATING THE GROWTH OF JACOBI FIELDS 225
9.2 WHEN DO LOCAL DIFFEOMORPHISMS BECOME COVERING MAPS? ... 231 9.3 SOME
CONSEQUENCES OF THE COVERING HOMOTOPY THEOREM 235 9.4 THE
CARTAN-HADAMARD THEOREM 238 9.5 PRELUDE TO RAUCH S THEOREM 240 9.5 A.
TRANSPLANTING VECTOR FIELDS 240 9.5 B. A SECOND BASIC PROPERTY OF THE
INDEX FORM 241 9.5 C. FLAG CURVATURE VERSUS CONJUGATE POINTS 243 9.6
RAUCH S FIRST COMPARISON THEOREM 244 9.7 JACOBI FIELDS ON SPACE FORMS
251 9.8 APPLICATIONS OF RAUCH S THEOREM 253 * REFERENCES FOR CHAPTER 9
256 CONTENTS XIX PART THREE SPECIAL FINSLER SPACES OVER THE REALS 257
CHAPTER 10 BERWALD SPACES AND SZABO S THEOREM FOR BERWALD SURFACES 257
10.0 PROLOGUE 257 10.1 BERWALD SPACES 258 10.2 VARIOUS CHARACTERIZATIONS
OF BERWALD SPACES 263 10.3 EXAMPLES OF BERWALD SPACES 266 10.4 A FACT
ABOUT FLAT LINEAR CONNECTIONS 272 10.5 CHARACTERIZING LOCALLY MINKOWSKI
SPACES BY CURVATURE 275 10.6 SZABO S RIGIDITY THEOREM FOR BERWALD
SURFACES 276 10.6 A. THE THEOREM AND ITS PROOF 276 10.6 B.
DISTINGUISHING BETWEEN Y-LOCAL AND Y-GLOBAL 279 * REFERENCES FOR CHAPTER
10 280 CHAPTER 11 RANDERS SPACES AND AN ELEGANT THEOREM 281 11.0 THE
IMPORTANCE OF RANDERS SPACES 281 11.1 RANDERS SPACES, POSITIVITY, AND
STRONG CONVEXITY 283 11.2 A MATRIX RESULT AND ITS CONSEQUENCES 287 11.3
THE GEODESIC SPRAY COEFFICIENTS OF A RANDERS METRIC 293 11.4 THE
NONLINEAR CONNECTION FOR RANDERS SPACES 298 11.5 A USEFUL AND ELEGANT
THEOREM 301 11.6 THE CONSTRUCTION OF Y-GLOBAL BERWALD SPACES 304 11.6 A.
THE ALGORITHM 304 11.6 B. AN EXPLICIT EXAMPLE IN THREE DIMENSIONS 306 *
REFERENCES FOR CHAPTER 11 309 CHAPTER 12 CONSTANT FLAG CURVATURE SPACES
AND AKBAR-ZADEH S THEOREM 311 12.0 PROLOGUE 311 12.1 CHARACTERIZATIONS
OF CONSTANT FLAG CURVATURE 312 12.2 USEFUL INTERPRETATIONS OF E AND E
314 12.3 GROWTH RATES OF SOLUTIONS OF E + A E = 0 320 12.4 AKBAR-ZADEH S
RIGIDITY THEOREM 325 12.5 FORMULAS FOR MACHINE COMPUTATIONS OF K 329
12.5 A. THE GEODESIC SPRAY COEFFICIENTS 329 12.5 B. THE PREDECESSOR OF
THE FLAG CURVATURE 330 XX CONTENTS 12.5 C. MAPLE CODES FOR THE GAUSSIAN
CURVATURE 331 12.6 A POINCARE DISC THAT IS ONLY FORWARD COMPLETE 333
12.6 A. THE EXAMPLE AND ITS YASUDA-SHIMADA PEDIGREE 334 12.6 B. THE
FINSLER FUNCTION AND ITS GAUSSIAN CURVATURE .... 335 12.6 C. GEODESIES;
FORWARD AND BACKWARD METRIC DISCS 336 12.6 D. CONSISTENCY WITH
AKBAR-ZADEH S RIGIDITY THEOREM ... 341 12.7 NON-RIEMANNIAN PROJECTIVELY
FLAT S 2 WITH K * 343 12.7 A. BRYANT S 2-PARAMETER FAMILY OF FINSLER
STRUCTURES ... 343 12.7 B. A SPECIFIC FINSLER METRIC FROM THAT FAMILY
345 * REFERENCES FOR CHAPTER 12 350 CHAPTER 13 RIEMANNIAN MANIFOLDS AND
TWO OF HOPF S THEOREMS 351 13.1 THE LEVI-CIVITA (CHRISTOFFEL) CONNECTION
351 13.2 CURVATURE 354 13.2 A. SYMMETRIES, BIANCHI IDENTITIES, THE RICCI
IDENTITY .... 354 13.2 B. SECTIONAL CURVATURE 355 13.2 C. RICCI
CURVATURE AND EINSTEIN METRICS 357 13.3 WARPED PRODUCTS AND RIEMANNIAN
SPACE FORMS 361 13.3 A. ONE SPECIAL CLASS OF WARPED PRODUCTS 361 13.3 B.
SPHERES AND SPACES OF CONSTANT CURVATURE 364 13.3 C. STANDARD MODELS OF
RIEMANNIAN SPACE FORMS 366 13.4 HOPF S CLASSIFICATION OF RIEMANNIAN
SPACE FORMS 369 13.5 THE DIVERGENCE LEMMA AND HOPF S THEOREM 376 13.6
THE WEITZENBOCK FORMULA AND THE BOCHNER TECHNIQUE 378 * REFERENCES FOR
CHAPTER 13 382 CHAPTER 14 MINKOWSKI SPACES, THE THEOREMS OF DEICKE AND
BRICKELL 383 14.1 GENERALITIES AND EXAMPLES 383 14.2 THE RIEMANNIAN
CURVATURE OF EACH MINKOWSKI SPACE 387 14.3 THE RIEMANNIAN LAPLACIAN IN
SPHERICAL COORDINATES 390 14.4 DEICKE S THEOREM 393 14.5 THE EXTRINSIC
CURVATURE OF THE LEVEL SPHERES OF F 397 14.6 THE GAUSS EQUATIONS 399
14.7 THE BLASCHKE-SANTALO INEQUALITY 403 14.8 THE LEGENDRE
TRANSFORMATION 406 14.9 A MIXED-VOLUME INEQUALITY, AND BRICKELL S
THEOREM 412 * REFERENCES FOR CHAPTER 14 418 BIBLIOGRAPHY 419 INDEX 427
|
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| dewey-ones | 516 - Geometry |
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| dewey-sort | 3516.373 |
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| id | DE-604.BV013248607 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:42:27Z |
| institution | BVB |
| isbn | 038798948X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009029724 |
| oclc_num | 247798899 |
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| physical | XX, 431 S. graph. Darst. : 24 cm |
| publishDate | 2000 |
| publishDateSearch | 2000 |
| publishDateSort | 2000 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Bao, David Verfasser aut An introduction to Riemann-Finsler geometry D. Bao ; S.-S. Chern ; Z. Shen New York [u.a.] Springer 2000 XX, 431 S. graph. Darst. : 24 cm txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 200 Literaturangaben Finsler-Geometrie Finsler spaces Geometry, Riemannian Finsler-Geometrie (DE-588)4451048-2 gnd rswk-swf Finsler-Geometrie (DE-588)4451048-2 s DE-604 Chern, Shiing-shen 1911-2004 Verfasser (DE-588)118520350 aut Shen, Zhongmin Verfasser aut Graduate texts in mathematics 200 (DE-604)BV000000067 200 HEBIS Datenaustausch Darmstadt application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009029724&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bao, David Chern, Shiing-shen 1911-2004 Shen, Zhongmin An introduction to Riemann-Finsler geometry Graduate texts in mathematics Finsler-Geometrie Finsler spaces Geometry, Riemannian Finsler-Geometrie (DE-588)4451048-2 gnd |
| subject_GND | (DE-588)4451048-2 |
| title | An introduction to Riemann-Finsler geometry |
| title_auth | An introduction to Riemann-Finsler geometry |
| title_exact_search | An introduction to Riemann-Finsler geometry |
| title_full | An introduction to Riemann-Finsler geometry D. Bao ; S.-S. Chern ; Z. Shen |
| title_fullStr | An introduction to Riemann-Finsler geometry D. Bao ; S.-S. Chern ; Z. Shen |
| title_full_unstemmed | An introduction to Riemann-Finsler geometry D. Bao ; S.-S. Chern ; Z. Shen |
| title_short | An introduction to Riemann-Finsler geometry |
| title_sort | an introduction to riemann finsler geometry |
| topic | Finsler-Geometrie Finsler spaces Geometry, Riemannian Finsler-Geometrie (DE-588)4451048-2 gnd |
| topic_facet | Finsler-Geometrie Finsler spaces Geometry, Riemannian |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009029724&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT baodavid anintroductiontoriemannfinslergeometry AT chernshiingshen anintroductiontoriemannfinslergeometry AT shenzhongmin anintroductiontoriemannfinslergeometry |