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CHAPTER ⅠTensor analysis1

1． Transformation of co？rdinates. The summation convention1

2． Contravariant vectors. Congruences of curves3

3． Invariants. Covariant vectors6

4． Tensors. Symmetric and skew-symmetric tensors9

5． Addition, subtraction and multiplication of tensors. Contraction12

6． Conjugate symmetric tensors of the second order. Associate tensors14

7． The Christoffel 3-index symbols and their relations17

8． Riemann symbols and the Riemaun tensor. The Ricci tensor19

9． Quadratic differential forms22

10． The equivalence of symmetric quadratic differential forms23

11． Covariant differentiation with respect to a tensor g？26

CHAPTER Ⅱ Introduction of a metric34

12． Definition of a metric. The fundamental tensor34

13．Angle of two vectors. Orthogonality37

14． Differential parameters. The normals to a hypersurface41

15． N-tuply orthogonal systems of hypersurfaces in a V？43

16． Metric properties of a space V？ immersed in a V？44

17． Geodesics48

18． Riemannian, normal and geodesic co？rdinates53

19． Geodesic form of the linear element. Finite equations of geodesics.57

20． Curvature of a curve60

21． Parallelism62

22． Parallel displacement and the Riemann tensor65

23． Fields of parallel vectors67

24． Associate directions. Parallelism in a sub-space72

25． Curvature of V？ at a point79

26． The Bianchi identity. The theorem of Schur82

27． Isometric correspondence of spaces of constant curvature. Motions in a V？84

28． Conformal spaces. Spaces conformal to a flat space89

CHAPTER Ⅲ Orthogonal ennuples96

29． Determination of tensors by means of the components of an orthogonal ennuple and invariants96

30． Coefficients of rotation. Geodesic congruences97

31． Determinants and matrices101

32． The orthogonal ennuple of Schmidt. As？ociate directions of higher orders. The Frenet formulas for a curve in a Vn103

33． Principal directions determined by a symmetric covariant tensor of the second order107

34． Geometrical interpretation of the Ricci tensor. The Ricci principal directions113

35． Condition that a congruence of an orthogonal ennuple be normal114

36． N-tuply orthogonal systems of hypersurfaces117

37． N-tuply orthogonal systems of hypersurfaces in a space conformal to a flat space119

38． Congruences canonical with respect to a given congruence125

39． Spaces for which the equations of geodesics admit a first integral128

40． Spaces with corresponding geodesics131

41． Certain spaces with corresponding geodesics135

CHAPTERⅣ The geometry of sub-spaces143

42． The normals to a space Vn immersed in a space Vm143

43． The Gauss and Codazzi equations for a hypersurface146

44． Curvature of a curve in a hypersurface150

45． Principal normal curvatures of a hypersurface and lines of curvature.152

46． Properties of the second fundamental form. Conjugate directions.Asymptotic directions155

47． Equations of Gauss and Codazzi for a Vn immersed in a Vm159

48． Normal and relative curvatures of a curve in a Vn immersed in a Vm164

49． The second fundamental form of a Vn in a Vm. Conjugate and asymp-totic directions166

50． Lines of curvature and mean curvature167

51． The fundamental equations of a Vn in a Vm in terms of invariants and an orthogonal ennuple170

52． Minimal varieties176

53． Hypersurfaces with indeterminate lines of curvature179

54． Totally geodesic varieties in a space183

CHAPTER Ⅴ Sub-spaces of a flat space187

55． The class of a space Vn187

56． A space Vn of class p＞l189

57． Evolutes of a Vn in an Sn+p192

58． A subspace Vn of a Vm immersed in an Sm+p195

59． Spaces Vn of class one197

60． Applicability of hypersurfaces of a flat space200

61． Spsces of constant curvature which are hypersurfaces of a flat space201

62． Coǒrdinates of Weierstrass. Motion in a space of constant curvature204

63． Equations of geodesics in a space of constant curvature in terms of coǒrdinates of Weierstrass207

64． Equations of a space Vn immersed in a Vm of constant curvature210

65． Spaces Vn conformal to an Sn214

CHAPTER Ⅵ Groups of motions221

66． Properties of continuous groups221

67． Transitive and intransitive groups. Invariant varieties225

68． Infinitesimal transformations which preserve geodesics227

69． Infinitesimal conformal transformations230

70． Infinitesimal motions. The equations of Killing233

71． Conditions of integrability of the equations of Killing. Spaces of constant curvature237

72． Infinitesimal translations239

73． Geometrical properties of the paths of a motion240

74． Spaces V？ which admit a group of motions241

75． Intransitive groups of motions244

76． Spaces V？ admitting a G？ of motions. Complete groups of motions of order n(n+1)/2-1245

77． Simply transitive groups as groups of motions247