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

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.

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.

• Do you know, or can you guess any reasonable condition under which such thing exists? – Mohammad Farajzadeh-Tehrani Apr 13 '10 at 22:27
• because in the very degenerate case where Reeb flow gives an S^1 action such things exists, So I was hopeful may be in some non-degenerate cases we can have it too. – Mohammad Farajzadeh-Tehrani Apr 13 '10 at 22:41
• Mohammad, I do not know an answer on that question, I'll try to think. May be one should look on examples.. – Petya Apr 13 '10 at 23:14

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.