On a Compact Lie Group Acting on a Manifold

Let M be a manifold (= connected, separable, locally euclidean space) of dimension n + 1, and G a compact connected Lie group acting on M in such a way that there is at least one n-dimensional orbit.2 In this paper, we show that the space of orbits M/G is homeomorphic to one of (i) a circle, (ii) an open interval, (iii) a half-open interval, or (iv) a closed interval. Moreover, there exists a subgroup N C G such that in (i) and (ii) M is homeomorphic to (G/N) X (M/G). In (iii) there is a subgroup K D N such that K/N is an r-sphere for some 0 < r < n, and such that M is homeomorphic to (G/N) X (M/G) with G/N identified to G/K over the end point. In (iv), a similar situation holds for subgroups K1 and K2 so that K1/N and K2/N are spheres (not necessarily of the same dimension). This, then, completely determines all manifolds admitting such a transformation group, and, moreover, the group G must act on M in the obvious way. (Thus, the operation of G is equivalent to G acting differentiably). In particular, this will show that the only two-dimensional manifolds which admit a compact connected Lie group operating non-trivially are as follows: (1) the torus, (2) the infinite circular cylinder, (3) the plane, (4) the (open) Moebius strip, (5) the sphere, (6) the Klein bottle, and (7) the projective plane: that is, precisely the Klein spaces. Each of these admits a circle group of operators, and in fact only the circle group operates effectively and non-transitively (except, of course, the trivial-i.e., one point-Lie group). A further application determines all compact, connected and effective Lie groups which can operate on a 3-dimensional manifold. All those 3-dimensional manifolds which admit a compact Lie group acting in such a way that there is a 2-dimensional orbit are determined. There are just 15 such. The author wishes to express his thanks to Dr. P. Conner, Profs. D. Montgomery, M. Goto, and A. L. Shields for their interest and suggestions.