Looking for almost complex structure on a contact manifold invariant under flow of Reeb vector field !?

Question

Notations : Suppose V is a closed contact compact manifold with contact form $\alpha$, of dimension 2n+1. Consider the symplectic sub-bundle $ \xi \subset TV $ given by $ \xi=$ ker($\alpha$). So $ \xi \rightarrow V $ is a vector bundle of rank $2n$ admitting the symplectic structure $d\alpha$. Also consider the Reeb vector field "$R$" determined by $\alpha$ and corresponding flow $\phi_t$. Assume that there is a finite number of simple periodic orbits of Reeb flow and all of them are non-degenerate.

Here is the question : In general there are plenty of almost complex structure on the symplectic bundle $\xi$ but is there any one which is invariant under Reeb flow, $\phi_t$ ?

I believe the answer will be no in general due to complicated behavior of $\phi_t$, but I wish that I am not right.

Answer 1

In general there is no invariant complex structure.

Let $\gamma$ be a closed orbit of the Reeb field. Consider a linearization $A$ of the Poincare return map along $\gamma$. $A$ is not, in general, a realification of a complex operator (with respect to any arbitrary complex structure). For example, as far as I remember, its Lefschetz number det(1−A) could be negative, which is impossible for a realification of a complex operator.

Answer 2

Mohammad,

When there is an invariant almost complex structure on $\xi\subset V$, then $V$ has a metric contact structure called a ``K-contact'' structure. Specifically, the metric is $$ g=\frac{1}{2}d\eta(\cdot,J\cdot) +\eta\otimes\eta$$.

There are contact structures on $S^2 \times T^{2n-1}$ for which it is easy to see there is no metric preserved by the Reeb action. (I think the example is in a paper of S. Tolman on toric contact manifolds.)

If one assumes further that $(\xi,J)$ is an integrable CR-structure, then the above metric is Sasaki. This case is much more thoroughly studied.