Riemannian geometry in an orthogonal frame

Contents......Page 12
Foreword......Page 6
Translator's Introduction......Page 8
Preface to the Russian Edition......Page 10
PRELIMINARIES......Page 19
1. Components of an infinitesimal displacement......Page 21
2. Relations among 1-forms of an orthonormal frame......Page 22
4. Moving frames......Page 23
5. Line element of the space......Page 24
6. Contravariant and covariant components......Page 25
7. Infinitesimal affine transformations of a frame......Page 26
8. Differentiation in a given direction......Page 27
9. Bilinear covariant of Frobenius......Page 28
10. Skew-symmetric bilinear forms......Page 31
11. Exterior quadratic forms......Page 32
12. Converse theorems. Cartan's Lemma......Page 33
13. Exterior differential......Page 36
14. Integral manifold of a system......Page 37
15. Necessary condition of complete integrability......Page 38
16. Necessary and sufficient condition of complete integrability of a system of Pfaffian equations......Page 39
17. Path independence of the solution......Page 40
18. Reduction of the problem of integration of a completely integrable system to the integration of a Cauchy system......Page 42
20. Relation between exterior differentials and the Stokes formula......Page 43
21. Orientation......Page 45
22. Exterior differential forms of arbitrary order......Page 47
24. The Gauss formula......Page 49
25. Generalization of Theorem 6 of No. 12......Page 51
A. GEOMETRY OF EUCLIDEAN SPACE......Page 53
26. Family of oblique trihedrons......Page 55
27. The family of orthonormal tetrahedrons......Page 56
28. Family of oblique trihedrons with a given line element......Page 57
29. Integration of system (I) by the method of the form invariance......Page 58
30. Particular cases......Page 59
31. Spaces of trihedrons......Page 61
32. The rigidity of the point space......Page 63
33. Geometric meaning of the Weyl theorem......Page 64
34. Deformation of the tangential space......Page 65
35. Deformation of the plane considered as a locus of straight lines......Page 68
36. Ruled space......Page 70
37. Transformation of the space with preservation of a line element......Page 73
38. Equivalence of reduction of a line element to a sum of squares to the choosing of a frame to be orthogonal......Page 76
39. Congruence and symmetry......Page 77
40. Determination of forms wji for given forms wi......Page 78
41. Three-dimensional case......Page 79
42. Absolute differentiation......Page 80
43. Divergence of a vector......Page 82
44. Differential parameters......Page 83
45. Notion of a tensor......Page 85
46. Tensor algebra......Page 87
47. Geometric meaning of a skew-symmetric tensor......Page 89
48. Scalar product of a bivector and a vector and of two bivectors......Page 91
49. Simple rotation of a rigid body around a point......Page 92
50. Absolute differentiation......Page 93
51. Rules of absolute differentiation......Page 94
52. Exterior differential tensor-valued form......Page 95
53. A problem of absolute exterior differentiation......Page 96
B. THE THEORY OF RIEMANNIAN MANIFOLDS......Page 99
54. The general notion of a manifold......Page 101
56. Riemannian manifolds. Regular metric......Page 102
58. Examples......Page 105
60. Locally compact manifold......Page 107
61. The holonomy group......Page 108
62. Discontinuity of the holonomy group of the locally Euclidean manifold......Page 109
63. Euclidean tangent metric......Page 111
64. Tangent Euclidean space......Page 112
65. The main notions of vector analysis......Page 114
66. Three methods of introducing a connection......Page 116
67. Euclidean metric osculating at a point......Page 117
68. Absolute differentiation of vectors on a Riemannian manifold......Page 119
69. Geodesics of a Riemannian manifold......Page 120
70. Generalization of the Frenet formulas. Curvature and torsion......Page 121
71. The theory of curvature of surfaces in a Riemannian manifold......Page 122
72. Geodesic torsion. The Enneper theorem......Page 124
74. The Dupin theorem on a triply orthogonal system......Page 126
75. Development of a Riemannian manifold in Euclidean space along a curve......Page 129
76. The constructed representation and the osculating Euclidean space......Page 130
77. Geodesics. Parallel surfaces......Page 131
78. Geodesics on a surface......Page 133
C. CURVATURE AND TORSION OF A MANIFOLD......Page 135
79. Determination of forms wji for given forms wi......Page 137
80. Condition of invariance of line element......Page 139
81. Axioms of equipollence of vectors......Page 141
82. Space with Euclidean connection......Page 142
83. Euclidean space of conjugacy......Page 143
84. Absolute exterior differential......Page 144
85. Torsion of the manifold......Page 145
86. Structure equations of a space with Euclidean connection......Page 146
87. Translation and rotation associated with a cycle......Page 147
89. Theorem of preservation of curvature and torsion......Page 148
90. The Bianchi identities in a Riemannian manifold......Page 151
91. The Riemann-Christoffel tensor......Page 152
92. Riemannian curvature......Page 154
93. The case n = 2......Page 155
94. The case n = 3......Page 157
95. Geometric theory of curvature of a three-dimensional Riemannian manifold......Page 158
96. Schur's theorem......Page 159
97. Example of a Riemannian space of constant curvature......Page 160
98. Determination of the Riemann-Christoffel tensor for a Riemannian curvature given for all planar directions......Page 162
99. Isotropic n-dimensional manifold......Page 163
101. Riemannian curvature in a direction of arbitrary dimension......Page 164
102. Ricci tensor. Einstein's quadric......Page 165
103. Congruence of spaces of the same constant curvature......Page 167
104. Existence of spaces of constant curvature......Page 169
105. Proof of Schur......Page 170
106. The system is satisfied by the solution constructed......Page 171
107. Spaces of constant positive curvature......Page 175
108. Mapping onto an n-dimensional projective space......Page 177
109. Hyperbolic space......Page 178
111. Geodesics in Riemannian manifold......Page 179
112. Pseudoequipollent vectors: pseudoparallelism......Page 180
113. Geodesics in spaces of constant curvature......Page 182
114. The Cayley metric......Page 184
D. THE THEORY OF GEODESIC LINES......Page 185
115. The field of geodesics......Page 187
116. Stationary state of the arc length of a geodesic in the family of lines joining two points......Page 188
117. The first variation of the arc length of a geodesic......Page 189
118. The second variation of the arc length of a geodesic......Page 190
119. The minimum for the arc length of a geodesic (Darboux's proof)......Page 191
120. Family of geodesics of equal length intersecting the same geodesic at a constant angle......Page 192
121. Distance between neighboring geodesics and curvature of a manifold......Page 197
122. The sum of the angles of a parallelogramoid......Page 199
123. Stability of a motion of a material system without external forces......Page 200
124. Investigation of the maximum and minimum for the length of a geodesic in the case Aij = const.......Page 201
125. Symmetric vectors......Page 203
126. Parallel transport by symmetry......Page 204
127. Determination of three-dimensional manifolds in which the parallel transport preserves the curvature......Page 205
128. Geodesic surface at a point. Severi's method of parallel transport of a vector......Page 207
129. Totally geodesic surfaces......Page 208
131. The Ricci theorem on orthogonal trajectories of totally geodesic surfaces......Page 209
E. EMBEDDED MANIFOLDS......Page 211
132. The Frenet formulas in a Riemannian manifold......Page 213
133. Determination of a curve with given curvature and torsion. Zero torsion curves in a space of constant curvature......Page 214
134. Curves with zero torsion and constant curvature in a space of constant negative curvature......Page 217
135. Integration of Frenet's equations of these curves......Page 221
136. Euclidean space of conjugacy......Page 223
137. The curvature of a Riemannian manifold occurs only in infinitesimals of second order......Page 224
138. The first two structure equations and their geometric meaning......Page 229
139. The third structure equation. Invariant forms (scalar and exterior)......Page 230
140. The second fundamental form of a surface......Page 231
141. Asymptotic lines. Euler's theorem. Total and mean curvature of a surface......Page 233
143. Geometric meaning of the form w......Page 234
144. Geodesic lines on a surface. Geodesic torsion. Enneper's theorem......Page 235
145. Laguerre's form......Page 239
146. Darboux's form......Page 240
147. Riemannian curvature of the ambient manifold......Page 242
148. The second group of structure equations......Page 243
149. Generalization of classical theorems on normal curvature and geodesic torsion......Page 244
150. Surfaces with a given line element in Euclidean space......Page 246
151. Problems on Laguerre's form......Page 247
152. Invariance of normal curvature under parallel transport of a vector......Page 249
153. Surfaces in a space of constant curvature......Page 251
154. Absolute variation of a tangent vector. Inner differentiation. Euler's curvature......Page 257
155. Tensor character of Euler's curvature......Page 258
156. The second system of structure equations......Page 259
157. Principal directions and principal curvatures......Page 260
158. Hypersurface in the Euclidean space......Page 261
159. Ellipse of normal curvature......Page 262
160. Generalization of classical notions......Page 265
161. Minimal surfaces......Page 266
162. Finding minimal surfaces......Page 267
Subject Index......Page 271