《An Introduction to Riemannian Geometry and The Tensor Calculus》

Chapter ⅠSOME PRELIMINARIES1

1．Determinants．Summation convention1

2．Differentiation of a determinant3

3．Matrices．Rank of a matrix4

4．Linear equations．Cramcr＇s rule4

5．Linear transformations6

6．Functional determinants7

7．Functional matrices9

8．Quadratic forms10

9．Real quadratic forms11

10．Pairs of quadratic forms12

11．Quadratic differential forms13

12．Differential equations14

EXAMPLES Ⅰ16

Chapter ⅡCOORDINATES．VECTORS．TENSORS18

13．Space of N dimensions．Subspaces．Directions at a point18

14．Transformations of coordinates．Contravariant vectors19

15．Scalar invariants．Covariant vectors21

16．Scalar product of two vectors23

17．Tensors of the second order24

18．Tensors of any order26

19．Symmetric and skew-symmetric tensors27

20．Addition and multiplication of tensors28

21．Contraction．Composition of tensors．Quotient law29

22．Reciprocal symmetric tensors of the second order31

EXAMPLES Ⅱ31

Chapter ⅢRIEMANNIAN METRIC35

23．Riemannian space．Fundamental tensor35

24．Length of a curve．Magnitude of a vector37

25．Associate covariant and contravariant vectors38

26．Inclination of two vectors．Orthogonal vectors39

27．Coordinate hypersurfaces．Coordinate curves40

28．Field of normals to a hypersurface42

29．N-ply orthogonal system of hypersurfaces44

30．Congruences of curves．Orthogonal ennuples45

31．Principal directions for a symmetric covariant tensor of the second order47

32．Euclidean space of n dimensions50

EXAMPLES Ⅲ53

Chapter ⅣCHRISTOFFEL＇S THREE-INDEX SYMBOLS．COVARIANT DIFFERENTIATION55

33．The Christoffel symbols55

34．Second derivatives of the x＇s with respect to the ？＇s56

35．Covariant derivative of a covariant vector．Curl of a vector58

36．Covariant derivative of a contravariant vector60

37．Derived vector in a given direction61

38．Covariant differentiation of tensors62

39．Covariant differentiation of sums and products64

40．Divergence of a vector65

41．Laplacian of a scalar invariant67

EXAMPLES Ⅳ68

Chapter ⅤCURVATURE OF A CURVE．GEODESICS．PARALLELISM OF VECTORS72

42．Curvature of a curve．Principal normal72

43．Geodesics．Euler＇s conditions73

44．Differential equations of geodesics75

45．Geodesic coordinates76

46．Riemannian coordinates79

47．Geodesic form of the linear element80

48．Geodesics in Euclidean space．Straight lines83

49．Parallel displacement of a vector of constant magnitude85

50．Parallelism for a vector of variable magnitude87

51．Subspaces of a Riemannian manifold89

52．Parallelism in a subspace91

53．Tendency and divergence of vectors with respect to subspace or enveloping space93

EXAMPLES Ⅴ95

Chapter ⅥCONGRUENCES AND ORTHOGONAL ENNUPLES98

54．Ricci＇s coefficients of rotation98

55．Curvature of a congruence．Geodesic congruences99

56．Commutation formula for the second derivatives along the arcs of the ennuple100

57．Reason for the name＂Coefficients of Rotation＂101

58．Conditions that a congruence be normal102

59．Curl of a congruence104

60．Congruences canonical with respect to a given congruence105

EXAMPLES Ⅵ109

Chapter ⅦRIEMANN SYMBOLS．CURVATURE OF A RIEMANNIAN SPACE110

61．Curvature tensor and Ricci tensor110

62．Covariant curvature tensor111

63．The identity of Bianchi113

64．Riemannian curvature of a Vn113

65．Formula for Riemannian curvature116

66．Theorem of Schur117

67．Mean curvature of a space for a given direction118

EXAMPLES Ⅶ121

Chapter ⅧHYPERSURFACES123

68．Notation．Unit normal123

69．Generalisod covariant differentiation124

70．Gauss＇s formulae．Second fundamental form126

71．Curvature of a curve in a hypersurface．Normal curvature128

72．Generalisation of Dupin＇s theorem130

73．Principal normal curvatures．Lines of curvature132

74．Conjugate directions and asymptotic directions in a hypersurface133

75．Tensor derivative of the unit normal．Derived vector135

76．The equations of Gauss and Codazzi138

77．Hypersurfaces with indeterminate lines of curvature．Totally geodesic hypersurfaces139

78．Family of hypersurfaces139

EXAMPLES Ⅷ141

Chapter ⅨHYPERSURFACES IN EUCLIDEAN SPACE．SPACES OF CONSTANT CURVATURE143

Euclidean Space143

79．Hyperplanes143

80．Hyperspheres144

81．Central quadric hypersurfaces146

82．Reciprocal quadrie hypersurfaces148

83．Conjugate radii149

84．An application151

85．Any hypersurface in Euclidean space152

86．Riemannian curvature．Ricci principal directions153

87．Evolute of a hypersurface in Euclidean space155

Spaces of Constant Curvature156

88．Riemannian curvature of a hypersphere156

89．Geodesics in a space of positive constant curvature158

EXAMPLES Ⅸ159

Chapter ⅩSUBSPACES OF A RIEMANNIAN SPACE162

90．Unit normals．Gauss＇s formulae162

91．Change from one set of normals to another163

92．Curvature of a curve in a subspace164

93．Conjugate and asymptotic directions in a subspace166

94．Generalisation of Dupin＇s theorem167

95．Derived vector of a unit normal169

96．Lines of curvature for a given normal171

EXAMPLES Ⅹ171

HISTORICAL NOTE173

BIBLIOGRAPHY180

INDEX188