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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