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