On a problem of Montgomery

Abstract

1. Let M be a manifold and let G be a compact Lie group acting on M. For xeM, G. denotes the isotropic subgroup of G at x, that is, the subgroup consisting of all the elements of G which leave x fixed. Then the following problem has been raised by Montgomery (see Ann. of Math. (2) vol. 50 (1949) p. 260): Do there exist a finite number of subgroups of G such that each Gx is conjugate to one of them? The purpose of this note is to show that, if M is compact and differentiability is assumed, then the answer is in the positive. Whether this result remains true without the assumption of differentiability is still unsettled. But if M is not compact, it is known that even for the analytic case the answer is in the negative. We shall give below an example to this effect, which was communicated to the author by Professor Montgomery.