Chapter Ⅰ SOME PRELIMINARIES 1
1．Determinants．Summation convention 1
2．Differentiation of a determinant 3
3．Matrices．Rank of a matrix 4
4．Linear equations．Cramcr＇s rule 4
5．Linear transformations 6
6．Functional determinants 7
7．Functional matrices 9
8．Quadratic forms 10
9．Real quadratic forms 11
10．Pairs of quadratic forms 12
11．Quadratic differential forms 13
12．Differential equations 14
EXAMPLES Ⅰ 16
Chapter Ⅱ COORDINATES．VECTORS．TENSORS 18
13．Space of N dimensions．Subspaces．Directions at a point 18
14．Transformations of coordinates．Contravariant vectors 19
15．Scalar invariants．Covariant vectors 21
16．Scalar product of two vectors 23
17．Tensors of the second order 24
18．Tensors of any order 26
19．Symmetric and skew-symmetric tensors 27
20．Addition and multiplication of tensors 28
21．Contraction．Composition of tensors．Quotient law 29
22．Reciprocal symmetric tensors of the second order 31
EXAMPLES Ⅱ 31
Chapter Ⅲ RIEMANNIAN METRIC 35
23．Riemannian space．Fundamental tensor 35
24．Length of a curve．Magnitude of a vector 37
25．Associate covariant and contravariant vectors 38
26．Inclination of two vectors．Orthogonal vectors 39
27．Coordinate hypersurfaces．Coordinate curves 40
28．Field of normals to a hypersurface 42
29．N-ply orthogonal system of hypersurfaces 44
30．Congruences of curves．Orthogonal ennuples 45
31．Principal directions for a symmetric covariant tensor of the second order 47
32．Euclidean space of n dimensions 50
EXAMPLES Ⅲ 53
Chapter Ⅳ CHRISTOFFEL＇S THREE-INDEX SYMBOLS．COVARIANT DIFFERENTIATION 55
33．The Christoffel symbols 55
34．Second derivatives of the x＇s with respect to the ？＇s 56
35．Covariant derivative of a covariant vector．Curl of a vector 58
36．Covariant derivative of a contravariant vector 60
37．Derived vector in a given direction 61
38．Covariant differentiation of tensors 62
39．Covariant differentiation of sums and products 64
40．Divergence of a vector 65
41．Laplacian of a scalar invariant 67
EXAMPLES Ⅳ 68
Chapter Ⅴ CURVATURE OF A CURVE．GEODESICS．PARALLELISM OF VECTORS 72
42．Curvature of a curve．Principal normal 72
43．Geodesics．Euler＇s conditions 73
44．Differential equations of geodesics 75
45．Geodesic coordinates 76
46．Riemannian coordinates 79
47．Geodesic form of the linear element 80
48．Geodesics in Euclidean space．Straight lines 83
49．Parallel displacement of a vector of constant magnitude 85
50．Parallelism for a vector of variable magnitude 87
51．Subspaces of a Riemannian manifold 89
52．Parallelism in a subspace 91
53．Tendency and divergence of vectors with respect to subspace or enveloping space 93
EXAMPLES Ⅴ 95
Chapter Ⅵ CONGRUENCES AND ORTHOGONAL ENNUPLES 98
54．Ricci＇s coefficients of rotation 98
55．Curvature of a congruence．Geodesic congruences 99
56．Commutation formula for the second derivatives along the arcs of the ennuple 100
57．Reason for the name＂Coefficients of Rotation＂ 101
58．Conditions that a congruence be normal 102
59．Curl of a congruence 104
60．Congruences canonical with respect to a given congruence 105
EXAMPLES Ⅵ 109
Chapter Ⅶ RIEMANN SYMBOLS．CURVATURE OF A RIEMANNIAN SPACE 110
61．Curvature tensor and Ricci tensor 110
62．Covariant curvature tensor 111
63．The identity of Bianchi 113
64．Riemannian curvature of a Vn 113
65．Formula for Riemannian curvature 116
66．Theorem of Schur 117
67．Mean curvature of a space for a given direction 118
EXAMPLES Ⅶ 121
Chapter Ⅷ HYPERSURFACES 123
68．Notation．Unit normal 123
69．Generalisod covariant differentiation 124
70．Gauss＇s formulae．Second fundamental form 126
71．Curvature of a curve in a hypersurface．Normal curvature 128
72．Generalisation of Dupin＇s theorem 130
73．Principal normal curvatures．Lines of curvature 132
74．Conjugate directions and asymptotic directions in a hypersurface 133
75．Tensor derivative of the unit normal．Derived vector 135
76．The equations of Gauss and Codazzi 138
77．Hypersurfaces with indeterminate lines of curvature．Totally geodesic hypersurfaces 139
78．Family of hypersurfaces 139
EXAMPLES Ⅷ 141
Chapter Ⅸ HYPERSURFACES IN EUCLIDEAN SPACE．SPACES OF CONSTANT CURVATURE 143
Euclidean Space 143
79．Hyperplanes 143
80．Hyperspheres 144
81．Central quadric hypersurfaces 146
82．Reciprocal quadrie hypersurfaces 148
83．Conjugate radii 149
84．An application 151
85．Any hypersurface in Euclidean space 152
86．Riemannian curvature．Ricci principal directions 153
87．Evolute of a hypersurface in Euclidean space 155
Spaces of Constant Curvature 156
88．Riemannian curvature of a hypersphere 156
89．Geodesics in a space of positive constant curvature 158
EXAMPLES Ⅸ 159
Chapter Ⅹ SUBSPACES OF A RIEMANNIAN SPACE 162
90．Unit normals．Gauss＇s formulae 162
91．Change from one set of normals to another 163
92．Curvature of a curve in a subspace 164
93．Conjugate and asymptotic directions in a subspace 166
94．Generalisation of Dupin＇s theorem 167
95．Derived vector of a unit normal 169
96．Lines of curvature for a given normal 171
EXAMPLES Ⅹ 171
HISTORICAL NOTE 173
BIBLIOGRAPHY 180
INDEX 188