UC Riverside

I have considered two main questions in my research. First, which foliations on a manifold are compatible with a particular symmetry group. Second, which contact structures are preserved by which symmetry groups. The existence of both foliations and contact structures have been studied for over a half-century with no consideration for symmetry groups. I have looked thus far at finite symmetry groups. Historically, the study of manifolds and symmetry groups goes back to Riemann's formulation of the manifold concept in his 1854 lecture on the foundations of geometry.

An equivalence class of smooth 1-forms L on a manifold M consists of all hL where h is any smooth function M to R. Given a smooth finite group action G on M, preserving this equivalence class means that there is a homomorphism E from G to {-1,1} and a L' in [L] such that for any g in G, g*L' = E(g)L'. In other words, there is a representative form L' such that g*L'=+-L'.

There are mostly negative results for orientation reversing group actions. Let M be a smooth oriented (4n-1)-manifold. If Z_2k, k>1 acts smoothly but not orientation preservingly on M then there is no contact form on M which is compatible with this group action. For foliations, there are no compatible codimension 1 foliations on S^3 with a group action containing an orientation reversing subgroup that is isomorphic to Z_2k, for k>1.

An equivalent condition for a smooth n-manifold M to have a G-invariant codimension 1 foliation is that each connected component of each fixed point set has Euler characteristic zero, where G is an odd order group that acts smoothly on M, with isotropy groups linearly ordered by inclusion.

The last result is constructive. Let M be a closed smooth oriented 3-manifold with a smooth orientation-preserving G-action, where G is a group of prime order p. Then there is a G-invariant contact form T on M. This form is constructed from an open-book decomposition, and a branched covering.