Kleinian Transformation Geometry

In this article we shall give a modern interpretation of transformation geometry. This subject has recently become of great interest to mathematics educators for use in kindergarten to high school, but has been paid too little attention at the college level. The usual approach to transformation geometry [5] or [12] consists of giving the classical geometries and then presenting their transformation groups. This certainly runs contrary to the ideas of the founder of transformation geometry, Felix Klein (1849-1929), who believed very strongly in a unified approach to mathematics whenever possible. I shall adopt the view that the proper approach to transformation geometry is through the geometry of homogeneous spaces and that this affords a modern interpretation of Klein's program (the Erlanger Programm) as outlined in his original paper of 1872 [7]. Roughly speaking, Klein's program says that a geometry on a space determines a group of transformations of the space and that a group of transformations on the space determines a geometry. While most modern mathematicians have been using the first idea (that very valuable information can be gained by looking at the group of transformations which leave the geometry invariant) we have neglected the second one. I claim that the theory of homogeneous spaces affords us a modern interpretation of the second point of Klein's program. I am not claiming that Klein foresaw the theory of homogeneous spaces but rather that with the help of the works of Riemann and E. Cartan (1869-1951) we can make precise (via the theory of homogeneous spaces) the notion that an arbitrarily chosen group will determine a geometry. If we are to adopt Klein's approach to transformations, what definition are we to take for geometry? There is certainly no easy answer to this question but the approach of G. F. B. Riemann (1826-1866) is certainly the most modern in spirit and is the one I shall use here. Riemann's inaugural address [111 begins: "As is well known, geometry presupposes the concept of space, as well as assuming the basic principles for constructions in space. It gives only nominal definitions of these things, while their essential specifications appear in the form of axioms. The relationship between these presuppositions [the concept of space, and the basic properties of space] is left in the dark; we do not see whether, or to what extent, any connection between them is necessary, or a priori whether any connection between them is even possible." This means that we must first decide what "space" should be. In section 2 we define space to be a homogeneous space; that is, the quotient space of a topological group G by a subgroup L so that M = GIL. In section 3 we assume that G is a subgroup of the group of nonsingular matrices (that is, G and L are Lie groups). In this section we add some geometric structure to that of space, which we call a geometry on the homogeneous space, and so obtain the notion of "lines." Throughout the last two sections of the paper we stress the fact that this definition of geometry includes in it, as special cases, Euclidean, Spherical and Hyperbolic geometries. We treat these three special cases in detail. (I do not claim, however, that this is the most general definition of geometry.) The first section of this paper gives a brief historical background before studying homogeneous spaces and their geometry. Where do homogeneous spaces belong in the realm of mathematics? They are not just used as an interpretation of Klein's Erlanger program; they are also used to do function theory (harmonic analysis [6]), to serve as models in differential geometry [8], and are used in mathematical physics [2]. It is as Klein remarks in the notes he added (in 1893) to the original Erlangen address ([7], p. 244): "A model, whether constructed and observed or only vividly imagined, is for this geometry not a means to an end, but the subject itself."