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Any two symplectic forms on $\mathbb{R}^{2n}$ are in the same cohomology class. But the usual symplectic form on a ball of radius 1 in Darboux coordinates does not have the same volume as the usual sy …

If the symplectic form integrates to a nonzero quantity on a compact surface in your manifold, it is not exact. For example, on $M=S^2\times S^1\times [0,1]$ with symplectic form $dA_{S^2} + d\varthet …

I proved that smooth projective planes, in the generalized sense of the axiomatic theory of projective planes, which have dimension 4 are diffeomorphic to the complex projective plane, using a general …

My first suggestion was: MR1066693 (91d:58073) Reviewed Guillemin, Victor(1-MIT); Sternberg, Shlomo(1-HRV) Symplectic techniques in physics. Second edition. Cambridge University Press, Cambridge, 19 …

Take a product of spheres, and let $\omega$ be half the area form on the first factor plus twice that on the second factor. The Chern classes of the tangent bundle are clearly invariant under switchin …

In cases when your Lie group is 1-dimensional and simple connected, i.e. the real number line, i.e. when there is precisely one function $J$ as the moment map, i.e. the cases you want to know about, t …

There is a time derivative implicit in forming the flow of a vector field. Writing out explicitly $X_H(p,q)=(H_q,-H_p)$, using subscripts for partial derivatives, the equations of flow lines of $X_H$ …

The issue is discussed, perhaps not completely clearly, in Henneaux and Teitelboim, Quantization of Gauge Systems. Princeton University Press, 1992. They prove, in chapter two, that the Dirac bracket …

Yes since functions which Poisson commute are constant on one another's Hamiltonian flows.

An $SU(n)$ structure is not a collection of charts whose transition maps have derivatives valued in $SU(n)$. It is a collection of bases of tangent spaces, forming a principal $SU(n)$-subbundle of the …

I never got further than visualizing pseudoholomorphic curves as just plane algebraic curves, drawing their real points, but then sometimes correcting a little by remembering their complex points as m …

There are infinitely many compactly supported symplectomorphisms of any symplectic manifold, which would then have to be represented by diffeomorphisms of $S$ preserving all marked conformal structure …

Suppose that $M \subset Z$ is a compact Lagrangian submanifold of a Kaehler manifold $Z$ with Kaehler form $\omega$. Take a function $f$ so that $\partial \bar\partial f=0$ at every point of $M$. Then …

It is not trivial. Its characteristic classes were worked out by Dmitrii Fuchs. In particular, the Maslov class is one of its nontrivial characteristic classes, and was the subject of a famous paper o …

The bundle $P$ is made out of frames, being a subbundle of the frame bundle $F$. So each point in $P$ is a basis of a tangent space of $M$. We can take any metric on $M$, and use it to parallel transp …