| Tensor analysis | |
| Transformation of coordinates. The summation convention | p. 1 |
| Contravariant vectors. Congruences of curves | p. 3 |
| Invariants. Covariant vectors | p. 6 |
| Tensors. Symmetric and skew-symmetric tensors | p. 9 |
| Addition, subtraction and multiplication of tensors. Contraction | p. 12 |
| Conjugate symmetric tensors of the second order. Associate tensors | p. 14 |
| The Christoffel 3-index symbols and their relations | p. 17 |
| Riemann symbols and the Riemaun tensor. The Ricci tensor | p. 19 |
| Quadratic differential forms | p. 22 |
| The equivalence of symmetric quadratic differential forms | p. 23 |
| Covariant differentiation with respect to a tensor g[subscript y] | p. 26 |
| Introduction of a metric | |
| Definition of a metric. The fundamental tensor | p. 34 |
| Angle of two vectors. Orthogonality | p. 37 |
| Differential parameters. The normals to a hypersurface | p. 41 |
| N-tuply orthogonal systems of hypersurfaces in a V[subscript n] | p. 43 |
| Metric properties of a space V[subscript n] immersed in a V[subscript m] | p. 44 |
| Geodesics | p. 48 |
| Riemannian, normal and geodesic coordinates | p. 53 |
| Geodesic form of the linear element. Finite equations of geodesics | p. 57 |
| Curvature of a curve | p. 60 |
| Parallelism | p. 62 |
| Parallel displacement and the Riemann tensor | p. 65 |
| Fields of parallel vectors | p. 67 |
| Associate directions. Parallelism in a sub-space | p. 72 |
| Curvature of v[subscript m] at a point | p. 79 |
| The Bianchi identity. The theorem of Schur | p. 82 |
| Isometric correspondence of spaces of constant curvature. Motions in a V[subscript n] | p. 84 |
| Conformal spaces. Spaces conformal to a flat space | p. 89 |
| Orthogonal ennuples | |
| Determination of tensors by means of the components of an orthogonal ennuple and invariants | p. 96 |
| Coefficients of rotation. Geodesic congruences | p. 97 |
| Determinants and matrices | p. 101 |
| The orthogonal ennuple of Schmidt. Associate directions of higher orders. The Frenet formulas for a curve in a V[subscript n] | p. 103 |
| Principal directions determined by a symmetric covariant tensor of the second order | p. 107 |
| Geometrical interpretation of the Ricci tensor. The Ricci principal directions | p. 113 |
| Condition that a congruence of an orthogonal ennuple be normal | p. 114 |
| N-tuply orthogonal systems of hypersurfaces | p. 117 |
| N-tuply orthogonal systems of hypersurfaces in a space conformal to a flat space | p. 119 |
| Congruences canonical with respect to a given congruence | p. 125 |
| Spaces for which the equations of geodesics admit a first integral | p. 128 |
| Spaces with corresponding geodesics | p. 131 |
| Certain spaces with corresponding geodesics | p. 135 |
| The geometry of sub-spaces | |
| The normals to a space V[subscript n] immersed in a space V[subscript m] | p. 143 |
| The Gauss and Codazzi equations for a hypersurface | p. 146 |
| Curvature of a curve in a hypersurface | p. 150 |
| Principal normal curvatures of a hypersurface and lines of curvature | p. 152 |
| Properties of the second fundamental form. Conjugate directions. Asymptotic directions | p. 155 |
| Equations of Gauss and Codazzi for a V[subscript n] immersed in a V[subscript m] | p. 159 |
| Normal and relative curvatures of a curve in a V[subscript n] immersed in a V[subscript m] | p. 164 |
| The second fundamental form of a V[subscript n] in a V[subscript m]. Conjugate and asymptotic directions | p. 166 |
| Lines of curvature and mean curvature | p. 167 |
| The fundamental equations of a V[subscript n] in a V[subscript m] in terms of invariants and an orthogonal ennuple | p. 170 |
| Minimal varieties | p. 176 |
| Hypersurfaces with indeterminate lines of curvature | p. 179 |
| Totally geodesic varieties in a space | p. 183 |
| Sub-spaces of a flat space | |
| The class of a space V[subscript n] | p. 187 |
| A space V[subscript n] of class p ] 1 | p. 189 |
| Evolutes of a V[subscript n] in an S[subscript n+p] | p. 192 |
| A subspace V[subscript n] of a V[subscript m] immersed in an S[subscript m+a] | p. 195 |
| Spaces V[subscript n] of class one | p. 197 |
| Applicability of hypersurfaces of a flat space | p. 200 |
| Spaces of constant curvature which are hypersurfaces of a flat space | p. 201 |
| Coordinates of Weierstrass. Motion in a space of constant curvature | p. 204 |
| Equations of geodesics in a space of constant curvature in terms of coordinates of Weierstrass | p. 207 |
| Equations of a space V[subscript n] immersed in a V[subscript m] of constant curvature | p. 210 |
| Spaces V[subscript n] conformal to an S[subscript n] | p. 214 |
| Groups of motions | |
| Properties of continuous groups | p. 221 |
| Transitive and intransitive groups. Invariant varieties | p. 225 |
| Infinitesimal transformations which preserve geodesics | p. 227 |
| Infinitesimal conformal transformations | p. 230 |
| Infinitesimal motions. The equations of Killing | p. 233 |
| Conditions of integrability of the equations of Killing. Spaces of constant curvature | p. 237 |
| Infinitesimal translations | p. 239 |
| Geometrical properties of the paths of a motion | p. 240 |
| Spaces V[subscript 2] which admit a group of motions | p. 241 |
| Intransitive groups of motions | p. 244 |
| Spaces V[subscript 3] admitting a G[subscript 2] of motions. Complete groups of motions of order n(n + 1)/2--1 | p. 245 |
| Simply transitive groups as groups of motions | p. 247 |
| Bibliography | p. 252 |
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