From a general point of view in developing the basis of differential geometry it is useful to consider the concept of a group and its connection to geometry.
It can be shown that by assuming the mappings:
T: x' i = F i (x1 , x2 , x3)
are one-to-one and defined on the whole space R 3 i.e. mappings of the whole space R 3 onto itself we can expect:
x' i = F i (x1 , x2 , x3), x' j = F' i (x'1 , x'2 , x'3) (i, j = 1,2,3........)
To be effected successively, thereby enabling us to obtain a composite mapping (or product) of the mappings such that:
x' j = {F j (x1 , x2 , x3), F2 (x1 , x2 , x3), F3 (x1 , x2 , x3) }
Thus a set G of mappings: T: x' i = F i (x1 , x2 , x3)
Is called a group of mappings or a transformation group if G has the following properties:
i) The identical mapping x'i = x i (i= 1,2,3......) is contained in G
ii) The inverse T -1 mapping of any mapping T contained in G is also an element of G
iii) For any arbitrary pair of mappings contained in G the product of these mappings is also an element of G
Then the direct congruent transformations form a group. Related to the above there are three axioms of equivalence:
1) Every geometric configuration is equivalent to itself
2) If a configuration A is equivalent to a configuration B then B is also equivalent to A.
3) If a configuration A is equivalent to a configuration B and B is also equivalent to configuration C then A is also equivalent to C.
Of particular importance are invariants (denoted I) corresponding to certain geometric objects which will remain unaffected by any mapping contained in G . Then I is said to be an invariant of the group G . For example the distance:
d = Ö å 3 b =1 ( y i - x i ) 2
Between two points ( y i ) and ( x i) of R 3 is an invariant with respect to the group of displacements. If related to some motion it will in general cause a change in the value of the coordinates, but d remains the same value. From the preceding we can consider some geometry which will be identical with a class of invariants for a certain transformation group. For example, affine geometry corresponds to the group of affine transformations:
x i = å 3 k =1 a ik x k + b i det (a ik ) ≠ 0
Of projective geometry to the group of projective transformations:
x i = å 3 b =1 b ik x k + b ik / å 3 b =1 a ik x k + b i
det (b ik ) ≠ 0
Now, since vector components are invariant with respect to translations, the basic transformation shown in Part 1 (see conditions (1), (2) of that post) corresponds to a transformation of the components of a vector which is of the form:
x' i = å 3 k =1 a ik x k , å 3 i =1 a ik a il = d kl ,
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