Transformation Groups of Spheres

Abstract

1. The compact Lie group G is said to be a transformation group of the space TW or to act on the space W if the following conditions are satisfied: a) to every element g of G there is associated a homeomorphism g(x) [x in TV] of W onto itself. b) if gl and g2 are elements of G then g9g2(X)] = (glg2)(X). c) the point g(x) depends continuously on the pair (g, x). Conditions a) and b) imply that to the identity element of G is associated the identity homeomorphism. The group G is said to act transitively if in addition to a), b), and c) the following fourth condition is satisfied: d) for any two points x and y of TW there is an element g in G such that g(x) = y. When d) is satisfied we say that TV is a homogeneous space under G. In this paper we take for W the n-dimensional sphere S' and study the question of what compact connected Lie groups can act transitively and effectively, (see 2 a) below), on S'. In I we prove a theorem on the structure of such a group which shows us that our main concern in the study of this problem is with simple groups. In II we study the question for simple groups using the Killing-Cartan classification, and we find that in general only those simple groups can be transitive and effective on SL which are well known to be so. In III we use our methods to draw some conclusions about the structure of certain subgroups of the rotation group of the n-dimensional sphere which we denote by Rn . Otherwise expressed Rn is the group of orthogonal transformations of determinant 1 on n + 1 real variables.