Symplectic manifold with totally geodesic foliation - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T22:53:57Z http://mathoverflow.net/feeds/question/111433 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/111433/symplectic-manifold-with-totally-geodesic-foliation Symplectic manifold with totally geodesic foliation Helge 2012-11-04T06:21:32Z 2012-11-04T06:21:32Z <p>Hallo,</p> <p>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.</p> <p>Helge</p>