Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question

2 Answers 2

up vote 7 down vote accepted

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.

share|improve this answer
    
Do you know, or can you guess any reasonable condition under which such thing exists? –  Mohammad F. 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 F. 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

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.