12. Transformation Groups

12. Transformation Groups 'We observed in Sect. 1 that the coordinate transformations of a simple coordinate manifold of class C" form a group whose elements, i.e. coordinate transformations, can be interpreted as (1,1) transformations of the arithmetic space of n dimensions into itself. By specializing this group, more precisely by limiting our attention to certain of its subgroups, we arrive a t various well known spaces, among these being the Euclidean metric space with which we shall be concerned in much of the following discussion in this book. In order to consider this problem in the proper perspective let us denote by G a set of (1,l) transformations of the arithmetic space of n dimensions into itself and let us write down the following axioms in which the undefined elements are points and Preferred coordinate systems. These axioms are : of

G,. Each preferred coordinate system is a (1,l) t r a n s f o r m a t i o ~ ~ the points into the arithmetic space of n dimensions,

G,. A n y transformation of coordinates from one preferred coordinate system to another belongs to G , G,. A n y coordinate system obtained from a preferred coordinate system b y a transformation belonging to G i s preferred,

G,.

There exists at least one preferred coordinate system.

Examining the consequences of the above axioms we see from G, that if P t)(xl,. . .,x") and P t ) ( y l , . . .,y") are two preferred coordinate systems there is a unique transformation f x l , . . . , x n ) t ) ( y l , . . .,y") relating the coordinates of these systems; it follows from G, that this transformation belongs to the set G. Also if we select any preferred coordinate system, the existence of which is specified by G,, it follows from the axioms G,, G, and G, 54

12. TRANSFORMATION GROUPS

55

that the preferred coordinate systems are those, and only those, obtainable from the selected system by transformations belonging to G. Finally it is easily seen that the set of transformations G constitutes a group. The underlying point set together with the set of preferred coordinate systems will be spoken of as a space. By the geometry of the space we mean the theory or body of theorems deducible from the above axioms G,, . . .G,. If the group G is the group of all ( 1 , l ) transformations of class C" of the n dimensional arithmetic space into itself, the space is identical with the simple coordinate manifold of class C" discussed in Sect. 1. We now consider several groups G which lead to well known mathematical spaces. These groups are defined by their representative transformations as follows z k = a tI x l + b ' ;

+ b'; 2 = a; x' + b'; X+ = a; X I

la;l

# 0,

(affine group),

(orthogonal group), (12.2)

a) a; = dlk, aza+ 1 k = pd,,;

(12.1)

p

> 0, (Euclidean group),

(12.3)

where there is a summation over the range 1 , . . . ,n on all repeated indices and the quantities dlk are the Kronecker symbols previously defined and denoted by S; in Sect. 1. The coefficients p, b' and a; in the above equations are constants. In the case of the orthogonal and Euclidean groups we see from the conditions imposed on the constants a; by the relations (12.2) and (12.3) that the determinant IaiI must be different from zero; however it must be assumed explicitly that the determinant la31 does not vanish in the case of the affine group in order for (12.1) to represent a ( 1 , l ) coordinate transformation. If G is the affine group the resulting space is called the affine space of n dimensions. We obtain the Euclidean metric space or the Euclidean space of n dimensions according as G is the orthogonal group or the Euclidean group. The preferred coordinate systems for the affine space are called Cartesian coordinate systems; the preferred coordinate systems are called rectangular Cartesian

56

12. TRANSFORMATION GROUPS

coordinate systems, or simply rectangular coordinate systems for brevity, in the Euclidean metric space and in the Euclidean space. By affine geometry we mean the theory of the affine space and by the Euclidean metric geometry and the Euclidean geometry we mean the theory of the Euclidean metric space and the Euclidean space respectively . As explained for the case of the simple coordinate manifold of class C" in Sect. 1 each of the above spaces will determine two oriented spaces for which the preferred coordinate systems are related by transformations whose functional determinants are positive. 'Thus an affine space determines two oriented affine spaces having preferred coordinate systems related by transformations (12.1) with iari > 0 ; these restricted transformations (12.1) obviously determine a group which will be called the pvoper affine group. Correspondingly the preferred coordinate systems of an oriented Euclidean metric space and an oriented Euclidean space will be related by transformations of the proper orthogonal group and the proper Euclidean group defined by (12.2) and (12.3) respec1 tively with Ia,"i > 0 ; we observe in this connection that IaiI = from the second set of relations (12.2) and hence I u , " ~ = 1 if (12.2) is to represent a proper orthogonal transformation. The affine, Euclidean metric and Euclidean spaces, as well as the oriented spaces which they determine, are characterized essentially by the group G of coordinate transformations relating their preferred coordinate systems. Interpreting the transformations of the group G as point transformations, rather than coordinate transformations in the strict sense, let us say that two configurations (sets of points) in the space are equivalent if one can be transformed into the other by a transformation of G. In particular equivalent configurations in an oriented Euclidean metric space are said to be congruent ; equivalent configurations are called similar in an oriented Euclidean space. Thus the ordinary concepts of congruence and similarity are given a precise meaning in terms of the transformations of a group.