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Tables of Contents for Riemannian Geometry
Chapter/Section Title
Page #
Page Count
CHAPTER I Tensor analysis
1
33
1. Transformation of coordinates. The summation convention
1
2
2. Contravariant vectors. Congruences of curves
3
3
3. Invariants. Covariant vectors
6
3
4. Tensors. Symmetric and skew-symmetric tensors
9
3
5. Addition, subtraction and multiplication of tensors. Contraction
12
2
6. Conjugate symmetric tensors of the second order. Associate tensors
14
3
7. The Christoffel 3-index symbols and their relations
17
2
8. Riemann symbols and the Riemann tensor. The Ricci tensor
19
3
9. Quadratic differential forms
22
1
10. The equivalence of symmetric quadratic differential forms
23
3
11. Covariant differentiation with respect to a tensor g
26
8
CHAPTER II Introduction of a metric
34
62
12. Definition of a metric. The fundamental tensor
34
3
13. Angle of two vectors. Orthogonality
37
4
14. Differential parameters. The normals to a hypersurface
41
2
15. N-tuply orthogonal systems of hypersurfaces in a V(n)
43
1
16. Metric properties of a space V(n) immersed in a V(m)
44
4
17. Geodesics
48
5
18. Riemannian, normal and geodesic coordinates
53
4
19. Geodesic form of the linear element. Finite equations of geodesics
57
3
20. Curvature of a curve
60
2
21. Parallelism
62
3
22. Parallel displacement and the Riemann tensor
65
2
23. Fields of parallel vectors
67
5
24. Associate directions. Parallelism in a sub-space
72
7
25. Curvature of V(n) at a point
79
3
26. The Bianchi identity. The theorem of Schur
82
2
27. Isometric correspondence of spaces of constant curvature. Motions in a V(n)
84
5
28. Conformal spaces. Spaces conformal to a flat space
89
7
CHAPTER III Orthogonal ennuples
96
47
29. Determination of tensors by means of the components of an orthogonal ennuple and invariants
96
1
30. Coefficients of rotation. Geodesic congruences
97
4
31. Determinants and matrices
101
2
32. The orthogonal ennuple of Schmidt. Associate directions of higher orders. The Frenet formulas for a curve in a V(n)
103
4
33. Principal directions determined by a symmetric covariant tensor of the second order
107
6
34. Geometrical interpretation of the Ricci tensor. The Ricci principal directions
113
1
35. Condition that a congruence of an orthogonal ennuple be normal
114
3
36. N-tuply orthogonal systems of hypersurfaces
117
2
37. N-tuply orthogonal systems of hypersurfaces in a space conformal to a flat space
119
6
38. Congruences canonical with respect to a given congruence
125
3
39. Spaces for which the equations of geodesics admit a first integral
128
3
40. Spaces with corresponding geodesics
131
4
41. Certain spaces with corresponding geodesics
135
8
CHAPTER IV The geometry of sub-spaces
143
44
42. The normals to a space V(n) immersed in a space V(m)
143
3
43. The Gauss and Codazzi equations for a hypersurface
146
4
44. Curvature of a curve in a hypersurface
150
2
45. Principal normal curvatures of a hypersurface and lines of curvature
152
3
46. Properties of the second fundamental form. Conjugate directions. Asymptotic directions
155
4
47. Equations of Gauss and Codazzi for a V(n) immersed in a V(m)
159
5
48. Normal and relative curvatures of a curve in a V(n) immersed in a V(m)
164
2
49. The second fundamental form of a V(n) in a V(m). Conjugate and asymptotic directions
166
1
50. Lines of curvature and mean curvature
167
3
51. The fundamental equations of a V(n) in a V(m) in terms of invariants and an orthogonal ennuple
170
6
52. Minimal varieties
176
3
53. Hypersurfaces with indeterminate lines of curvature
179
4
54. Totally geodesic varieties in a space
183
4
CHAPTER V Sub-spaces of a flat space
187
34
55. The class of a space V(n)
187
2
56. A space V(n) of class p greater than 1
189
3
57. Evolutes of a V(n) of a V(m) in an S(n+p)
192
3
58. A subspace V(n) of a V(m) immersed in an S(m+l)
195
2
59. Spaces V(n) of class one
197
3
60. Applicability of hypersurfaces of a flat space
200
1
61. Spaces of constant curvature which are hypersurfaces of a flat space
201
3
62. Coordinates of Weierstrass. Motion in a space of constant curvature
204
3
63. Equations of geodesics in a space of constant curvature in terms of coordinates of Weierstrass
207
3
64. Equations of a space V(n) immersed in a V(m) of constant curvature
210
4
65. Spaces V(n) conformal to an S(n)
214
7
CHAPTER VI Groups of motions
221
 
66. Properties of continuous groups
221
4
67. Transitive and intransitive groups. Invariant varieties
225
2
68. Infinitesimal transformations which preserve geodesics
227
3
69. Infinitesimal conformal transformations
230
3
70. Infinitesimal motions. The equations of Killing
233
4
71. Conditions of integrability of the equations of Killing. Spaces of constant curvature
237
2
72. Infinitesimal translations
239
1
73. Geometrical properties of the paths of a motion
240
1
74. Spaces V(2) which admit a group of motions
241
3
75. Intransitive groups of motions
244
1
76. Spaces V(2) admitting a G(2) of motions. Complete groups of motions of order n(n+1) 2-1
245
2
77. Simply transitive groups as groups of motions
247
5
Bibliography
252