One-parameter transformation groups of the three-sphere

Abstract

INTRODUCTION. In [3], Seifert describes all the fiberings of the three-sphere. The fibers are the orbits when the circle group acts effectively on the three-sphere without fixed points in a certain way. In [2,? 6.7], Montgomery and Zippin show that when the circle group acts on the three-sphere effectively but with fixed points its action is described in a suitable coordinate system as a rotation about an axis. In this paper the methods of [2] and of [3] are combined to show that the circle group can act effectively on the three-sphere without fixed points in no other way than in one of the ways described in [3 ]. The result is stated precisely in the following theorem. The author has had the benefit of frequient conversations with Deane Montgomery during the preparation of this paper. NOTATION. In the following pages 'R' denotes the set of all real numbers, 'S2' denotes the set of all triples x = (x1, X2, X3) of real numbers satisfying xl2+x2+x = 1, and 'S3' or 'S' denotes the set of all quadruples x = (xi, X2, X3, X4) of real numbers satisfying x2+x2+x2 +x41. A is the null set.