《Invariants of Quadratic Differential Forms》求取 ⇩

Ⅰ．FORMAL PRELIMINARIES1

1．The summation convention1

2，3．The Kronecker deltas3

4．Linear equations6

5，6．Functional Determinants7

7．Derivative of a determinant8

8．Numerical relations8

9，10．Minors，cofactors，and the Laplace expansion9

11．Historical11

Ⅱ．DIFFERENTIAL INVARIANTS13

1．N-dimensional space13

2．Transformations of coordinates13

3．Invariants14

4．Differential invariants15

5，6．Differentials and contravariant vectors16

7．A general class of invariants19

8．Tensors19

9．Relative scalars20

10．Covariant vectors21

11，12．Algebraic combinations of tensors22

13．The commonness of tensors24

14．Numerical tensors25

15．Combinations of vectors26

16．Historical and general remarks27

Ⅲ．QUADRATIC DIFFERENTIAL FORMS30

1．Differential forms30

2．Linear differential forms30

3，4．Quadratic differential forms31

5．Invariants derived from basic invariants32

6，7．Invariants of a quadratic differential form32

8，9．The fundamental affine connection33

10．Affine connections in general35

11，12．Covariant differentiation36

13．Geodesic coordinates38

14，15．Formulas of covariant differentiation39

16，17．The curvature tensor41

18，19．Riemann-Christoffel tensor43

20．Reduction theorems44

21．Historical remarks47

22．Scalar invariants48

Ⅳ．EUCLIDEAN GEOMETRY50

1，2．Euclidean geometry50

3．Euclidean affine geometry53

4，5．Euclidean vector analysis55

6．Associated vectors and tensors56

7．Distance and scalar product57

8，9．Area59

10．First order differential parameters60

11．Euclidean covariant differentiation61

12．The divergence62

13．The Laplacian or Lamé differential parameter of the second order63

14．The curl of a vector64

15，16．Generalized divergence and curl64

17．Historical remarks66

Ⅴ．THE EQUIVALENCE PROBLEM67

1．Riemannian geometry67

2．The theory of surfaces68

3．Spaces immersed in a Euclidean space69

4，5．Condition that a Riemannian space be Euclidean69

6．The equivalence problem72

7．A lemma on mixed systems73

8．Equivalence theorem for quadratic differential forms76

9．Equivalence of affine connections77

10，11．Automorphisms of a quadratic differential form78

12．Equivalence theorem in terms of scalars79

13．Historical remarks80

Ⅵ．NORMAL COORDINATES82

1，2．Affine geometry of paths82

3，4．Affine normal coordinates85

5．Affine extensions87

6．The affine normal tensors89

7，8．The replacement theorems90

9，10，11，12．The curvature tensor and the normal tensors91

13，14，15，16，17．Affine extensions of the fundamental tensor94

18．Historical and general remarks100

19．Formulas for the extensions of tensors102