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