Riemannian geometry in an orthogonal frame: from lectures delivered by Élie Cartan at the Sorbonne in 1926 - 27
Gespeichert in:
| 1. Verfasser: | |
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| Weitere Verfasser: | |
| Format: | Buch |
| Sprache: | English Russian |
| Veröffentlicht: |
River Edge, NJ
World Scientific
2001
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| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVII, 259 S. Ill. |
| ISBN: | 981024746X 9810247478 |
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| 240 | 1 | 0 | |a Rimanova geometrija v ortogonalnom repere |
| 245 | 1 | 0 | |a Riemannian geometry in an orthogonal frame |b from lectures delivered by Élie Cartan at the Sorbonne in 1926 - 27 |c Translated from Russian by Vladislav V. Goldberg |
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Datensatz im Suchindex
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| adam_text | RIEMANNIAN GEOMETRY IN AN ORTHOGONAL FRAME FROM LECTURES DELIVERED BY
ELIE CARTAN AT THE SORBONNE IN 1926-27 TRANSLATED FROM RUSSIAN BY
VLADISLAV V. GOLDBERG NEW JERSEY INSTITUTE OF TECHNOLOGY NEWARK, NEW
JERSEY, U.S.A. FOREWORD BY S. S. CHERN FE WORLD SCIENTIFIC *
NEWJERSEYLONDON »SINQAPORE* NEW JERSEY »LONDON * SINGAPORE * HONG KONG
CONTENTS FOREWORD V TRANSLATOR S INTRODUCTION VII PREFACE TO THE RUSSIAN
EDITION IX PRELIMINARIES 1 CHAPTER 1 METHOD OF MOVING FRAMES 3 1.
COMPONENTS OF AN INFINITESIMAL DISPLACEMENT 3 2. RELATIONS AMONG 1-FORMS
OF AN ORTHONORMAL FRAME 4 3. FINDING THE COMPONENTS OF A GIVEN FAMILY OF
TRIHEDRONS 5 4. MOVING FRAMES 5 5. LINE ELEMENT OF THE SPACE 6 6.
CONTRAVARIANT AND COVARIANT COMPONENTS 7 7. INFINITESIMAL AFFINE
TRANSFORMATIONS OF A FRAME 8 CHAPTER 2 THE THEORY OF PFAFFIAN FORMS 9 8.
DIFFERENTIATION IN A GIVEN DIRECTION 9 9. BILINEAR COVARIANT OF
FROBENIUS 10 10. SKEW-SYMMETRIC BILINEAR FORMS 13 11. EXTERIOR QUADRATIC
FORMS 14 12. CONVERSE THEOREMS. CARTAN S LEMMA 15 13. EXTERIOR
DIFFERENTIAL 18 CHAPTER 3 INTEGRATION OF SYSTEMS OF PFAFFIAN
DIFFERENTIAL EQUATIONS 19 14. INTEGRAL MANIFOLD OF A SYSTEM 19 15.
NECESSARY CONDITION OF COMPLETE INTEGRABILITY 20 16. NECESSARY AND
SUFFICIENT CONDITION OF COMPLETE INTEGRABILITY OF A SYSTEM OF PFAFFIAN
EQUATIONS 21 17. PATH INDEPENDENCE OF THE SOLUTION 22 XI XLL CONTENTS
18. REDUCTION OF THE PROBLEM OF INTEGRATION OF A COMPLETELY INTEGRABLE
SYSTEM TO THE INTEGRATION OF A CAUCHY SYSTEM 24 19. FIRST INTEGRALS OF A
COMPLETELY INTEGRABLE SYSTEM 25 20. RELATION BETWEEN EXTERIOR
DIFFERENTIALS AND THE STOKES FORMULA 25 21. ORIENTATION 27 CHAPTER 4
GENERALIZATION 29 22. EXTERIOR DIFFERENTIAL FORMS OF ARBITRARY ORDER 29
23. THE POINCARE THEOREM 31 24. THE GAUSS FORMULA 31 25. GENERALIZATION
OF THEOREM 6 OF NO. 12 33 A. GEOMETRY OF EUCLIDEAN SPACE 35 CHAPTER 5
THE EXISTENCE THEOREM FOR A FAMILY OF FRAMES WITH GIVEN INFINITESIMAL
COMPONENTS W L AND U) 1 - 37 26. FAMILY OF OBLIQUE TRIHEDRONS 37 27. THE
FAMILY OF ORTHONORMAL TETRAHEDRONS 38 28. FAMILY OF OBLIQUE TRIHEDRONS
WITH A GIVEN LINE ELEMENT 39 29. INTEGRATION OF SYSTEM (I) BY THE METHOD
OF THE FORM INVARIANCE 40 30. PARTICULAR CASES 41 31. SPACES OF
TRIHEDRONS 43 CHAPTER 6 THE FUNDAMENTAL THEOREM OF METRIE GEOMETRY 45
32. THE RIGIDITY OF THE POINT SPACE 45 33. GEOMETRIE MEANING OF THE WEYL
THEOREM 46 34. DEFORMATION OF THE TANGENTIAL SPACE 47 35. DEFORMATION OF
THE PLANE CONSIDERED AS A LOCUS OF STRAIGHT LINES 50 36. RULED SPACE 52
CHAPTER 7 VECTOR ANALYSIS IN AN N-DIMENSIONAL EUCLIDEAN SPACE 55 37.
TRANSFORMATION OF THE SPACE WITH PRESERVATION OF A LINE ELEMENT 55 38.
EQUIVALENCE OF REDUCTION OF A LINE ELEMENT TO A SUM OF SQUARES TO THE
CHOOSING OF A FRAME TO BE ORTHOGONAL 58 39. CONGRUENCE AND SYMMETRY 59
40. DETERMINATION OF FORMS U FOR GIVEN FORMS U 1 60 41.
THREE-DIMENSIONAL CASE 61 42. ABSOLUTE DIFFERENTIATION 62 43. DIVERGENCE
OF A VECTOR 64 CONTENTS XNI 44. DIFFERENTIAL PARAMETERS 65 CHAPTER 8 THE
FUNDAMENTAL PRINCIPLES OF TENSOR ALGEBRA 67 45. NOTION OF A TENSOR 67
46. TENSOR ALGEBRA 69 47. GEOMETRIE MEANING OF A SKEW-SYMMETRIC TENSOR
71 48. SCALAR PRODUET OF A BIVECTOR AND A VECTOR AND OF TWO BIVECTORS 73
49. SIMPLE ROTATION OF A RIGID BODY AROUND A POINT 74 CHAPTER 9 TENSOR
ANALYSIS 75 50. ABSOLUTE DIFFERENTIATION 75 51. RULES OF ABSOLUTE
DIFFERENTIATION 76 52. EXTERIOR DIFFERENTIAL TENSOR-VALUED FORM 77 53. A
PROBLEM OF ABSOLUTE EXTERIOR DIFFERENTIATION 78 B. THE THEORY OF
RIEMANNIAN MANIFOLDS 81 CHAPTER 10 THE NOTION OF A MANIFOLD 83 54. THE
GENERAL NOTION OF A MANIFOLD 83 55. ANALYTIC REPRESENTATION 84 56.
RIEMANNIAN MANIFOLDS. REGULAER METRIC 84 CHAPTER 11 LOCALLY EUCLIDEAN
RIEMANNIAN MANIFOLDS 87 57. DEFINITION OF A LOCALLY EUCLIDEAN MANIFOLD
87 58. EXAMPLES 87 59. RIEMANNIAN MANIFOLD WITH AN EVERYWHERE REGULAER
METRIC 89 60. LOCALLY COMPACT MANIFOLD 89 61. THE HOLONOMY GROUP 90 62.
DISCONTINUITY OF THE HOLONOMY GROUP OF THE LOCALLY EUCLIDEAN MANIFOLD 91
CHAPTER 12 EUCLIDEAN SPACE TANGENT AT A POINT 93 63. EUCLIDEAN TANGENT
METRIC 93 64. TANGENT EUCLIDEAN SPACE 94 65. THE MAIN NOTIONS OF VECTOR
ANALYSIS 96 66. THREE METHODS OF INTRODUCING A CONNECTION 98 67.
EUCLIDEAN METRIC OSCULATING AT A POINT 9 9 CHAPTER 13 OSCULATING
EUCLIDEAN SPACE 101 68. ABSOLUTE DIFFERENTIATION OF VECTORS ON A
RIEMANNIAN MANIFOLD 101 69. GEODESICS OF A RIEMANNIAN MANIFOLD 102 70.
GENERALIZATION OF THE FRENET FORMULAS. CURVATURE AND TORSION 103 71. THE
THEORY OF CURVATURE OF SURFACES IN A RIEMANNIAN MANIFOLD 104 72.
GEODESIC TORSION. THE ENNEPER THEOREM 106 73. CONJUGATE DIRECTIONS 108
XIV CONTENTS 74. THE DUPIN THEOREM ON A TRIPLY ORTHOGONAL SYSTEM 108
CHAPTER 14 EUCLIDEAN SPACE OF CONJUGACY ALONG A LINE 111 75. DEVELOPMENT
OF A RIEMANNIAN MANIFOLD IN EUCLIDEAN SPACE ALONG A CURVE 111 76. THE
CONSTRUCTED REPRESENTATION AND THE OSCULATING EUCLIDEAN SPACE 112 77.
GEODESICS. PARALLEL SURFACES 113 78. GEODESICS ON A SURFACE 115 C.
CURVATURE AND TORSION OF A MANIFOLD 117 CHAPTER 15 SPACE WITH A
EUCLIDEAN CONNECTION 119 79. DETERMINATION OF FORMS W? FOR GIVEN FORMS
U 1 119 80. CONDITION OF INVARIANCE OF LINE ELEMENT 121 81. AXIOMS OF
EQUIPOLLENCE OF VECTORS 123 82. SPACE WITH EUCLIDEAN CONNECTION 124 83.
EUCLIDEAN SPACE OF CONJUGACY 125 84. ABSOLUTE EXTERIOR DIFFERENTIAL 126
85. TORSION OF THE MANIFOLD 127 86. STRUCTURE EQUATIONS OF A SPACE WITH
EUCLIDEAN CONNECTION 128 87. TRANSLATION AND ROTATION ASSOCIATED WITH A
CYCLE 129 88. THE BIANCHI IDENTITIES 130 89. THEOREM OF PRESERVATION OF
CURVATURE AND TORSION 130 CHAPTER 16 RIEMANNIAN CURVATURE OF A MANIFOLD
133 90. THE BIANCHI IDENTITIES IN A RIEMANNIAN MANIFOLD 133 91. THE
RIEMANN-CHRISTOFFEL TENSOR 134 92. RIEMANNIAN CURVATURE 136 93. THE CASE
N = 2 137 94. THE CASE N = 3 139 95. GEOMETRIE THEORY OF CURVATURE OF A
THREE-DIMENSIONAL RIEMANNIAN MANIFOLD 140 96. SCHUR S THEOREM 14 1 97.
EXAMPLE OF A RIEMANNIAN SPACE OF CONSTANT CURVATURE 142 98.
DETERMINATION OF THE RIEMANN-CHRISTOFFEL TENSOR FOR A RIEMANNIAN
CURVATURE GIVEN FOR ALL PLANAR DIRECTIONS 144 99. ISOTROPIE
N-DIMENSIONAL MANIFOLD 145 100. CURVATURE IN TWO DIFFERENT
TWO-DIMENSIONAL PLANAR DIRECTIONS 146 101. RIEMANNIAN CURVATURE IN A
DIRECTION OF ARBITRARY DIMENSION 146 102. RICCI TENSOR. EINSTEIN S
QUADRIC 147 CHAPTER 17 SPACES OF CONSTANT CURVATURE 149 103. CONGRUENCE
OF SPACES OF THE SAME CONSTANT CURVATURE 149 104. EXISTENCE OF SPACES OF
CONSTANT CURVATURE 151 105. PROOFOF SCHUR 152 CONTENTS XV 106. THE
SYSTEM IS SATISFIED BY THE SOLUTION CONSTRUCTED 153 CHAPTER 18 GEOMETRIE
CONSTRUCTION OF A SPACE OF CONSTANT CURVATURE 157 107. SPACES OF
CONSTANT POSITIVE CURVATURE 157 108. MAPPING ONTO AN N-DIMENSIONAL
PROJEETIVE SPACE 159 109. HYPERBOLIC SPACE 16 0 110. REPRESENTATION OF
VECTORS IN HYPERBOLIC GEOMETRY 161 111. GEODESICS IN RIEMANNIAN MANIFOLD
161 112. PSEUDOEQUIPOLLENT VECTORS: PSEUDOPARALLELISM 162 113. GEODESICS
IN SPACES OF CONSTANT CURVATURE 164 114. THE CAYLEY METRIC 166 D. THE
THEORY OF GEODESIC LINES 167 CHAPTER 19 VARIATIONAL PROBLEMS FOR
GEODESICS 169 115. THE FIELD OF GEODESICS 169 116. STATIONARY STATE OF
THE ARC LENGTH OF A GEODESIC IN THE FAMILY OF LINES JOINING TWO POINTS
170 117. THE FIRST VARIATION OF THE ARC LENGTH OF A GEODESIC 171 118.
THE SECOND VARIATION OF THE ARC LENGTH OF A GEODESIC 172 119. THE
MINIMUM FOR THE ARC LENGTH OF A GEODESIC (DARBOUX S PROOF) 173 120.
FAMILY OF GEODESICS OF EQUAL LENGTH INTERSECTING THE SAME GEODESIC AT A
CONSTANT ANGLE 174 CHAPTER 20 DISTRIBUTION OF GEODESICS NEAR A GIVEN
GEODESIC 179 121. DISTANCE BETWEEN NEIGHBORING GEODESICS AND CURVATURE
OF A MANIFOLD 179 122. THE SUM OF THE ANGLES OF A PARALLELOGRAMOID 18 1
123. STABILITY OF A MOTION OF A MATERIAL SYSTEM WITHOUT EXTERNAL FORCES
182 124. INVESTIGATION OF THE MAXIMUM AND MINIMUM FOR THE LENGTH OF A
GEODESIC IN THE CASE AIJ = CONST. 183 125. SYMMETRIE VECTORS 185 126.
PARALLEL TRANSPORT BY SYMMETRY 186 127. DETERMINATION OF
THREE-DIMENSIONAL MANIFOLDS, IN WHICH THE PARALLEL TRANSPORT PRESERVES
THE CURVATURE 187 CHAPTER 21 GEODESIC SURFACES 189 128. GEODESIC SURFACE
AT A POINT. SEVERI S METHOD OF PARALLEL TRANSPORT OF A VECTOR 189 129.
TOTALLY GEODESIC SURFACES 190 130. DEVELOPMENT OF LINES OF A TOTALLY
GEODESIC SURFACE ON A PLANE 191 131. THE RICCI THEOREM ON ORTHOGONAL
TRAJEETORIES OF XVI CONTENTS TOTALLY GEODESIC SURFACES 191 E. EMBEDDED
MANIFOLDS 193 CHAPTER 22 LINES IN A RIEMANNIAN MANIFOLD 195 132. THE
FRENET FORMULAS IN A RIEMANNIAN MANIFOLD 195 133. DETERMINATION OF A
CURVE WITH GIVEN CURVATURE AND TORSION. ZERO TORSION CURVES IN A SPACE
OF CONSTANT CURVATURE 196 134. CURVES WITH ZERO TORSION AND CONSTANT
CURVATURE IN A SPACE OF CONSTANT NEGATIVE CURVATURE 199 135. INTEGRATION
OF FRENET S EQUATIONS OF THESE CURVES 203 136. EUCLIDEAN SPACE OF
CONJUGACY 205 137. THE CURVATURE OF A RIEMANNIAN MANIFOLD OCCURS ONLY IN
INFINITESIMALS OF SECOND ORDER 206 CHAPTER 23 SURFACES IN A
THREE-DIMENSIONAL 211 RIEMANNIAN MANIFOLD 138. THE FIRST TWO STRUCTURE
EQUATIONS AND THEIR GEOMETRIC MEANING 211 139. THE THIRD STRUCTURE
EQUATION. INVARIANT FORMS (SCALAR AND EXTERIOR) 212 140. THE SECOND
FUNDAMENTAL FORM OF A SURFACE 213 141. ASYMPTOTIC LINES. EULER S
THEOREM. TOTAL AND MEAN CURVATURE OF A SURFACE 215 142. CONJUGATE
TANGENTS 216 143. GEOMETRIC MEANING OF THE FORM IP 216 144. GEODESIC
LINES ON A SURFACE. GEODESIC TORSION. ENNEPER S THEOREM 217 CHAPTER 24
FORMS OF LAGUERRE AND DARBOUX 221 145. LAGUERRE S FORM 221 146.
DARBOUX S FORM 222 147. RIEMANNIAN CURVATURE OF THE AMBIENT MANIFOLD 224
148. THE SECOND GROUP OF STRUCTURE EQUATIONS 225 149. GENERALIZATION OF
CLASSICAL THEOREMS ON NORMAL CURVATURE AND GEODESIC TORSION 226 150.
SURFACES WITH A GIVEN LINE ELEMENT IN EUCLIDEAN SPACE 228 151. PROBLEMS
ON LAGUERRE S FORM 229 152. INVARIANCE OF NORMAL CURVATURE UNDER
PARALLEL TRANSPORT OF A VECTOR 231 153. SURFACES IN A SPACE OF CONSTANT
CURVATURE 233 CHAPTER 25 P-DIMENSIONAL SUBMANIFOLDS IN A RIEMANNIAN
MANIFOLD OF N DIMENSIONS 239 154. ABSOLUTE VARIATION OF A TANGENT
VECTOR. INNER DIFFERENTIATION. EULER S CURVATURE 239 155. TENSOR
CHARACTER OF EULER S CURVATURE 240 CONTENTS XVII 156. THE SECOND SYSTEM
OF STRUCTURE EQUATIONS 241 PARTICULAR CASES A HYPERSURFACE IN A
FOUR-DIMENSIONAL SPACE 157. PRINCIPAL DIRECTIONS AND PRINCIPAL
CURVATURES 242 158. HYPERSURFACE IN THE EUCLIDEAN SPACE 24 3 159.
ELLIPSE OF NORMAL CURVATURE 244 TWO-DIMENSIONAL SURFACES IN A
FOUR-DIMENSIONAL MANIFOLD 160. GENERALIZATION OF CLASSICAL NOTIONS 247
161. MINIMAL SURFACES 248 162. FINDING MINIMAL SURFACES 249 INDEX
|
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| id | DE-604.BV014439315 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:02:46Z |
| institution | BVB |
| isbn | 981024746X 9810247478 |
| language | English Russian |
| lccn | 2002277436 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009869477 |
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| physical | XVII, 259 S. Ill. |
| publishDate | 2001 |
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| spelling | Cartan, Elie 1869-1951 Verfasser (DE-588)119240424 aut Rimanova geometrija v ortogonalnom repere Riemannian geometry in an orthogonal frame from lectures delivered by Élie Cartan at the Sorbonne in 1926 - 27 Translated from Russian by Vladislav V. Goldberg River Edge, NJ World Scientific 2001 XVII, 259 S. Ill. txt rdacontent n rdamedia nc rdacarrier Geometry, Riemannian Riemannsche Geometrie (DE-588)4128462-8 gnd rswk-swf Riemannsche Geometrie (DE-588)4128462-8 s DE-604 Golʹdberg, Vladislav V. 1936- (DE-588)128639644 trl GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009869477&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Cartan, Elie 1869-1951 Riemannian geometry in an orthogonal frame from lectures delivered by Élie Cartan at the Sorbonne in 1926 - 27 Geometry, Riemannian Riemannsche Geometrie (DE-588)4128462-8 gnd |
| subject_GND | (DE-588)4128462-8 |
| title | Riemannian geometry in an orthogonal frame from lectures delivered by Élie Cartan at the Sorbonne in 1926 - 27 |
| title_alt | Rimanova geometrija v ortogonalnom repere |
| title_auth | Riemannian geometry in an orthogonal frame from lectures delivered by Élie Cartan at the Sorbonne in 1926 - 27 |
| title_exact_search | Riemannian geometry in an orthogonal frame from lectures delivered by Élie Cartan at the Sorbonne in 1926 - 27 |
| title_full | Riemannian geometry in an orthogonal frame from lectures delivered by Élie Cartan at the Sorbonne in 1926 - 27 Translated from Russian by Vladislav V. Goldberg |
| title_fullStr | Riemannian geometry in an orthogonal frame from lectures delivered by Élie Cartan at the Sorbonne in 1926 - 27 Translated from Russian by Vladislav V. Goldberg |
| title_full_unstemmed | Riemannian geometry in an orthogonal frame from lectures delivered by Élie Cartan at the Sorbonne in 1926 - 27 Translated from Russian by Vladislav V. Goldberg |
| title_short | Riemannian geometry in an orthogonal frame |
| title_sort | riemannian geometry in an orthogonal frame from lectures delivered by elie cartan at the sorbonne in 1926 27 |
| title_sub | from lectures delivered by Élie Cartan at the Sorbonne in 1926 - 27 |
| topic | Geometry, Riemannian Riemannsche Geometrie (DE-588)4128462-8 gnd |
| topic_facet | Geometry, Riemannian Riemannsche Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009869477&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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