The differential invariants of generalized spaces

CONTENTS ......Page 5
1. Space. Coordinates ......Page 10
2. Affine connection ......Page 14
3. Affine geometry of paths ......Page 16
4. Projective geometry of paths ......Page 19
5. Riemann or metric space ......Page 20
6. Space of distant parallelism ......Page 25
7. Conformal space ......Page 30
8. Weyl space. Gauge ......Page 32
9. Transformation theory of space ......Page 35
10. Tensors ......Page 40
11. Invariants ......Page 45
12. Parallel displacement of a vector around an infinitesimal closed circuit ......Page 48
13. Covariant differentiation ......Page 52
14. Alternative methods of covariant differentiation. Extension ......Page 57
15. Differential parameters ......Page 59
16. Affine representation of projective spaces ......Page 63
17. Some geometrical interpretations ......Page 66
18. Projective tensors and invariants ......Page 67
19. Transformations of the group *G ......Page 69
20. Fundamental conformal-affine tensor ......Page 76
21. The representation of conformal spaces ......Page 77
22. Conformal tensors and invariants ......Page 81
23. Completion of the incomplete covariant derivative. General case ......Page 84
24. An extension of the preceding method ......Page 86
25. Systems algebraically equivalent to the system of equations of transformation of the components of a conformal tensor ......Page 88
26. Exceptional case K = 0 ......Page 90
28. The complete conformal curvature tensor and its successive covariant derivatives ......Page 91
29. Affine normal coordinates ......Page 94
30. Absolute normal coordinates ......Page 97
31. Projective normal coordinates ......Page 101
32. General theory of extension ......Page 106
33. Some formulae of extension ......Page 108
34. Scalar differentiation in a space of distant parallelism ......Page 110
35. Differential invariants defined by means of normal coordinates. Normal tensors ......Page 112
36. A generalization of the affine normal tensors ......Page 115
37. Formulae of repeated extension ......Page 117
38. A theorem on the affine connection ......Page 118
39. Replacement theorems ......Page 119
41. Identities in the components of the normal tensors ......Page 122
42. Identities of the space of distant parallelism ......Page 125
43. Determination of the components of the normal tensors in terms of the components of their extensions ......Page 126
44. Generalization of the preceding identities ......Page 130
45. Space determination by tensor invariants ......Page 132
46. Relations between the components of the extensions of the normal tensors ......Page 133
47. Convergence proofs ......Page 136
48. Relations between the components of certain invariants of the space of distant parallelism ......Page 139
49. Determination of the components of the affine normal tensors in terms of the components of the curvature tensor and its covariant derivatives ......Page 140
50. Curvature. Theorem of Schur ......Page 142
51. Identities in the components of the projective curvature tensor ......Page 146
52. Certain divergence identities ......Page 148
53. A general method for obtaining divergence identities ......Page 150
54. Numbers of algebraically independent components of certain spatial invariants ......Page 153
55. Abstract groups ......Page 159
56. Finite continuous groups ......Page 160
57. Essential parameters ......Page 162
58. The parameter groups ......Page 163
59. Fundamental differential equations of an r-parameter group ......Page 164
60. Transformation theory connected with the fundamental differential equations ......Page 166
61. Equivalent r-parameter groups ......Page 167
62. Constants of composition ......Page 169
63. Group space and its structure ......Page 170
64. Infinitesimal transformations ......Page 171
65. Transitive and intransitive groups. Invariant sub-spaces ......Page 173
66. Invariant functions ......Page 174
67. Groups defined by the equations of transformation of the components of tensors ......Page 175
68. Infinitesimal transformations of the affine and metric groups ......Page 176
70. Absolute metric differential invariants of order zero ......Page 178
71. General theorems on the independence of the differential equations ......Page 179
72. Number of independent differential equations. Affine case ......Page 185
73. Number of independent differential equations. Metric case ......Page 187
74. Exceptional case of two dimensions ......Page 189
75. Fundamental sets of absolute scalar differential invariants ......Page 192
76. Rational invariants ......Page 194
77. Absolute scalar differential parameters ......Page 195
78. Independence of the differential equations of the differential parameters ......Page 196
79. Fundamental sets of differential parameters ......Page 202
80. Extension to relative tensor differential invariants ......Page 203
81. Equivalence of generalized spaces ......Page 207
82. Normal coordinates and the equivalence problem ......Page 210
84. A theorem on mixed systems of partial differential equations ......Page 212
85. Finite equivalence theorem for affinely connected spaces ......Page 214
86. Finite equivalence theorem for metric spaces ......Page 216
87. Finite equivalence theorem for spaces of distant parallelism ......Page 217
89. Equivalence of two dimensional conformal spaces ......Page 218
90. Finite equivalence theorem for conformal spaces of three or more dimensions ......Page 221
91. Spatial arithmetic invariants ......Page 222
92. Differential conditions of reducibility ......Page 225
93. Flat spaces ......Page 226
95. Algebraic conditions for the reducibility of the affine space of paths to a metric space ......Page 228
96. Algebraic conditions for the reducibility of the affine space of paths to a Weyl space ......Page 229
97. Regular systems of partial differential equations ......Page 231
98. Extension to tensor differential equations ......Page 234
100. Groups of independent components ......Page 236
101. Special case of two dimensions ......Page 238
102. General case of n (>= 3) dimensions ......Page 239
103. The existence theorems in normal coordinates ......Page 243
104. Convergence of the A series ......Page 246
105. Convergence of the g series ......Page 249
INDEX ......Page 251