Riemannian geometry in an orthogonal frame : from lectures delivered by Élie Cartan at the Sorbonne in 1926-1927 / translated from Russian by Vladislav V. Goldberg ; foreword by S. S. Chern.

Format:

Book

Author/Creator:

Cartan, Elie, 1869-1951.

Contributor:

Finikov, S. P. (Sergeĭ Pavlovich), 1883-1964.

Standardized Title:

Rimanova geometrii͡a v ortogonalʹnom repere. English

Language:

English

Russian

Subjects (All):

Geometry, Riemannian.

Physical Description:

xvii, 259 pages : illustrations ; 23 cm

Place of Publication:

River Edge, NJ : World Scientific, [2001]

Summary:

Elie Cartan's 1926-27 lectures on Riemannian geometry in an orthogonal frame were translated into Russian in book form in 1960. Now translated from Russian by Vladislav V. Goldberg, professor of mathematics at the New Jersey Institute of Technology and editor of the Russian edition, this book highlights Cartan's use of orthogonal frames to investigate the geometry of Riemannian manifolds and to solve problems in Euclidean and non-Euclidean spaces and in geodesics. The book introduces innovations such as the notion of intrinsic normal differentiation and the Gaussian torsion of a submanifold in a Euclidean multidimensional space or in a space of constant curvature. Annotation copyrighted by Book News, Inc., Portland, OR.

Contents:

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

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

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 [omega superscript i] and [omega superscript i subscript j] 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 Metric Geometry 45

32. The rigidity of the point space 45

33. Geometric 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 [omega superscript j subscript i] for given forms [omega superscript i] 60

41. Three-dimensional case 61

42. Absolute differentiation 62

43. Divergence of a vector 64

44. Differential parameters 65

Chapter 8 The Fundamental Principles of Tensor Algebra 67

45. Notion of a tensor 67

46. Tensor algebra 69

47. Geometric meaning of a skew-symmetric tensor 71

48. Scalar product 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. Regular metric 84

Chapter 11 Locally Euclidean Riemannian Manifolds 87

57. Definition of a locally Euclidean manifold 87

59. Riemannian manifold with an everywhere regular 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 99

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

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 [omega superscript j subscript i] for given forms [omega superscript i] 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. Geometric theory of curvature of a three-dimensional Riemannian manifold 140

96. Schur's theorem 141

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. Isotropic 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. Proof of Schur 152

106. The system is satisfied by the solution constructed 153

Chapter 18 Geometric Construction of a Space of Constant Curvature 157

107. Spaces of constant positive curvature 157

108. Mapping onto an n-dimensional projective space 159

109. Hyperbolic space 160

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 181

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 A[subscript ij] = const 183

125. Symmetric 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 trajectories of 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 Riemannian Manifold 211

138. The first two structure equations nad 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 [psi] 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

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 243

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.

Notes:

Translated from the 1960 Russian ed., which was translated and edited from original lecture notes by S.P. Finikov as, Rimanova geometriya v orthogonalʹnom repere.

Includes bibliographical references and index.

ISBN:

981024746X

9810247478

OCLC:

49356062