Written in the wake of the advent of Relativity by an author who made important contributions to projective and differential geometry, and topology, this early Cambridge Tract in Mathematics and Theoretical Physics aimed to assist students of the time from the fields of differential geometry and mathematical physics. Beginning by introducing formal preliminaries, the text continues, bringing the underlying differential invariant theory that to this day remains relevant in a range of geometrical and physical applications, to the fore.

Preface

v

Formal Preliminaries

The summation convention

1

(2)

The Kronecker deltas

3

(3)

Linear equations

6

(1)

Functional determinants

7

(1)

Derivative of a determinant

8

(1)

Numerical relations

8

(1)

Minors, cofactors, and the Laplace expansion

9

(2)

Historical

11

(2)

Differential Invariants

N-dimensional space

13

(1)

Transformations of coordinates

13

(1)

Invariants

14

(1)

Differential invariants

15

(1)

Differentials and contravariant vectors

16

(3)

A general class of invariants

19

(1)

Tensors

19

(1)

Relative scalars

20

(1)

Covariant vectors

21

(1)

Algebraic combinations of tensors

22

(2)

The commonness of tensors

24

(1)

Numerical tensors

25

(1)

Combinations of vectors

26

(1)

Historical and general remarks

27

(3)

Quadratic Differential Forms

Differential forms

30

(1)

Linear differential forms

30

(1)

Quadratic differential forms

31

(1)

Invariants derived from basic invariants

32

(1)

Invariants of a quadratic differential form

32

(1)

The fundamental affine connection

33

(2)

Affine connections in general

35

(1)

Covariant differentiation

36

(2)

Geodesic coordinates

38

(1)

Formulas of covariant differentiation

39

(2)

The curvature tensor

41

(2)

Riemann-Christoffel tensor

43

(1)

Reduction theorems

44

(3)

Historical remarks

47

(1)

Scalar invariants

48

(2)

Euclidean Geometry

Euclidean geometry

50

(3)

Euclidean affine geometry

53

(2)

Euclidean vector analysis

55

(1)

Associated vectors and tensors

56

(1)

Distance and scalar product

57

(2)

Area

59

(1)

First order differential parameters

60

(1)

Euclidean covariant differentiation

61

(1)

The divergence

62

(1)

The Laplacian or Lame differential parameter of the second order

63

(1)

The curl of a vector

64

(1)

Generalized divergence and curl

64

(2)

Historical remarks

66

(1)

The Equivalence Problem

Riemannian geometry

67

(1)

The theory of surfaces

68

(1)

Spaces immersed in a Euclidean space

69

(1)

Condition that a Riemannian space be Euclidean

69

(3)

The equivalence problem

72

(1)

A lemma on mixed systems

73

(3)

Equivalence theorem for quadratic differential forms

76

(1)

Equivalence of affine connections

77

(1)

Automorphisms of a quadratic differential form

78

(1)

Equivalence theorem in terms of scalars

79

(1)

Historical remarks

80

(2)

Normal Coordinates

Affine geometry of paths

82

(3)

Affine normal coordinates

85

(2)

Affine extensions

87

(2)

The affine normal tensors

89

(1)

The replacement theorems

90

(1)

The curvature tensor and the normal tensors

91

(3)

Affine extensions of the fundamental tensor

94

(6)

Historical and general remarks

100

(2)

Formulas for the extensions of tensors

102

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