On Smooth SL(m,R)×SL(n,R)-Actions on S^m+n-1*

Abstract

We shall study smooth SL(m,R)×SL(n,R) actions on S^m+n-1, each of which is an extension of the standard SO(m)×SO(n) action. We shall show such an action is characterized by a smooth R²-action on the circle. Moreover, we shall show there are uncountably many topologically distinct such extensions.