An Introduction to Riemannian Geometry and The Tensor CalculusPDF电子书下载

Chapter Ⅰ SOME PRELIMINARIES 1

1．Determinants．Summation convention 1

2．Differentiation of a determinant 3

3．Matrices．Rank of a matrix 4

4．Linear equations．Cramcr＇s rule 4

5．Linear transformations 6

6．Functional determinants 7

7．Functional matrices 9

8．Quadratic forms 10

9．Real quadratic forms 11

10．Pairs of quadratic forms 12

11．Quadratic differential forms 13

12．Differential equations 14

EXAMPLES Ⅰ 16

Chapter Ⅱ COORDINATES．VECTORS．TENSORS 18

13．Space of N dimensions．Subspaces．Directions at a point 18

14．Transformations of coordinates．Contravariant vectors 19

15．Scalar invariants．Covariant vectors 21

16．Scalar product of two vectors 23

17．Tensors of the second order 24

18．Tensors of any order 26

19．Symmetric and skew-symmetric tensors 27

20．Addition and multiplication of tensors 28

21．Contraction．Composition of tensors．Quotient law 29

22．Reciprocal symmetric tensors of the second order 31

EXAMPLES Ⅱ 31

Chapter Ⅲ RIEMANNIAN METRIC 35

23．Riemannian space．Fundamental tensor 35

24．Length of a curve．Magnitude of a vector 37

25．Associate covariant and contravariant vectors 38

26．Inclination of two vectors．Orthogonal vectors 39

27．Coordinate hypersurfaces．Coordinate curves 40

28．Field of normals to a hypersurface 42

29．N-ply orthogonal system of hypersurfaces 44

30．Congruences of curves．Orthogonal ennuples 45

31．Principal directions for a symmetric covariant tensor of the second order 47

32．Euclidean space of n dimensions 50

EXAMPLES Ⅲ 53

Chapter Ⅳ CHRISTOFFEL＇S THREE-INDEX SYMBOLS．COVARIANT DIFFERENTIATION 55

33．The Christoffel symbols 55

34．Second derivatives of the x＇s with respect to the ？＇s 56

35．Covariant derivative of a covariant vector．Curl of a vector 58

36．Covariant derivative of a contravariant vector 60

37．Derived vector in a given direction 61

38．Covariant differentiation of tensors 62

39．Covariant differentiation of sums and products 64

40．Divergence of a vector 65

41．Laplacian of a scalar invariant 67

EXAMPLES Ⅳ 68

Chapter Ⅴ CURVATURE OF A CURVE．GEODESICS．PARALLELISM OF VECTORS 72

42．Curvature of a curve．Principal normal 72

43．Geodesics．Euler＇s conditions 73

44．Differential equations of geodesics 75

45．Geodesic coordinates 76

46．Riemannian coordinates 79

47．Geodesic form of the linear element 80

48．Geodesics in Euclidean space．Straight lines 83

49．Parallel displacement of a vector of constant magnitude 85

50．Parallelism for a vector of variable magnitude 87

51．Subspaces of a Riemannian manifold 89

52．Parallelism in a subspace 91

53．Tendency and divergence of vectors with respect to subspace or enveloping space 93

EXAMPLES Ⅴ 95

Chapter Ⅵ CONGRUENCES AND ORTHOGONAL ENNUPLES 98

54．Ricci＇s coefficients of rotation 98

55．Curvature of a congruence．Geodesic congruences 99

56．Commutation formula for the second derivatives along the arcs of the ennuple 100

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

58．Conditions that a congruence be normal 102

59．Curl of a congruence 104

60．Congruences canonical with respect to a given congruence 105

EXAMPLES Ⅵ 109

Chapter Ⅶ RIEMANN SYMBOLS．CURVATURE OF A RIEMANNIAN SPACE 110

61．Curvature tensor and Ricci tensor 110

62．Covariant curvature tensor 111

63．The identity of Bianchi 113

64．Riemannian curvature of a Vn 113

65．Formula for Riemannian curvature 116

66．Theorem of Schur 117

67．Mean curvature of a space for a given direction 118

EXAMPLES Ⅶ 121

Chapter Ⅷ HYPERSURFACES 123

68．Notation．Unit normal 123

69．Generalisod covariant differentiation 124

70．Gauss＇s formulae．Second fundamental form 126

71．Curvature of a curve in a hypersurface．Normal curvature 128

72．Generalisation of Dupin＇s theorem 130

73．Principal normal curvatures．Lines of curvature 132

74．Conjugate directions and asymptotic directions in a hypersurface 133

75．Tensor derivative of the unit normal．Derived vector 135

76．The equations of Gauss and Codazzi 138

77．Hypersurfaces with indeterminate lines of curvature．Totally geodesic hypersurfaces 139

78．Family of hypersurfaces 139

EXAMPLES Ⅷ 141

Chapter Ⅸ HYPERSURFACES IN EUCLIDEAN SPACE．SPACES OF CONSTANT CURVATURE 143

Euclidean Space 143

79．Hyperplanes 143

80．Hyperspheres 144

81．Central quadric hypersurfaces 146

82．Reciprocal quadrie hypersurfaces 148

83．Conjugate radii 149

84．An application 151

85．Any hypersurface in Euclidean space 152

86．Riemannian curvature．Ricci principal directions 153

87．Evolute of a hypersurface in Euclidean space 155

Spaces of Constant Curvature 156

88．Riemannian curvature of a hypersphere 156

89．Geodesics in a space of positive constant curvature 158

EXAMPLES Ⅸ 159

Chapter Ⅹ SUBSPACES OF A RIEMANNIAN SPACE 162

90．Unit normals．Gauss＇s formulae 162

91．Change from one set of normals to another 163

92．Curvature of a curve in a subspace 164

93．Conjugate and asymptotic directions in a subspace 166

94．Generalisation of Dupin＇s theorem 167

95．Derived vector of a unit normal 169

96．Lines of curvature for a given normal 171

EXAMPLES Ⅹ 171

HISTORICAL NOTE 173

BIBLIOGRAPHY 180

INDEX 188