## Symplectic manifold with totally geodesic foliation

Hallo,

Let $(M,I,\omega)$ be a symplectic manifold. On this we can introduce a Poisson structure by $[A,B] = \omega(X_{A}, X_{B})$ where $X_{A}, X_{B}$ are defined by $\omega(X_{A}, *) = dA, \omega(X_{B}, *) = dB$ where $A,B$ are smooth functions on the manifold $M$. With respect to this structure there exists a foliation on $M$ by symplectic manifolds. Let assume here for simplicity that the foliation is regular and at least of codimension 2. My question is now: are the leaves of this foliation totally geodesic?if no, what assumptions needs one to make in order that the leaves are totally geodesic? I hope for a lot of answers. tanks in advance.

Helge

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Ho do you obtain the symplectic foliation of M? If you are referring to the singular foliation with symplectic leaves which is always associated to a Poisson structure, then in this case M is the unique leaf. – Paolo Ghiggini Nov 4 at 8:35
I said that the foliation should be at least of codimension 2. – Helge Nov 4 at 11:15
@Helge: You haven't defined (or mentioned) $I$, and maybe you are using this to define your foliation. As Paolo mentioned, the 'natural' symplectic leaves for the Poisson structure you defined are just the manifold $M$ itself, so you must be using something else to construct your foliation of codimension at least $2$. Moreover, you haven't mentioned where you are going to get a metric (or at least an affine or projective connection), and without that, the whole notion of 'geodesic' doesn't make sense. Maybe you can edit your question to explain this. As it is, there's nothing to answer. – Robert Bryant Nov 4 at 13:04
A theory from symplectic geometry states that a poisson manifold can be foliated by symplectic leaves. Now I suppose that I have a symplectic manifold. this is poisson and thus can be foliated by symplectic leaves. I also assume that this leaves are of codimension at least 2. this is an assumption. – Helge Nov 4 at 13:27
@helge. As paolo ghiggini and robert bryant said, if a poisson manifold is symplectic then its symplectic foliation consists of a single leaf. Therefore there is no example of a poisson manifold which is symplectic and have codimension 2 symplectic leaves. – DamienC Nov 4 at 14:39
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