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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 $X_H$ …

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 …

Yes. You reduce the structure group of a contact $(2n+1)$-manifold to those bases of the tangent space for which the first $2n$ vectors form a conformal symplectic basis for the contact hyperplanes. T …

V. I. Arnold, Mathematical Methods for Classical Mechanics, p. 280. L. D. Landau and E. M. Lifshitz, Mechanics, p. 157.

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 …

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 …

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 …

In any holomorphic chart, real analytic submanifolds remain real analytic. If your Lagrangian manifolds are not real analytic, they cannot become real analytic in holomorphic coordinates. In fact, you …

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 …

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 …

This follows from theorem 6.2 (and the first sentence in the proof) of Mario J. Micallef and Brian White, The structure of branch points in minimal surfaces and in pseudoholomorphic curves, Ann. of Ma …

Write the left invariant Maurer-Cartan 1-form $\omega$ on $SO(n)$ as $$ \begin{pmatrix} \omega^i_j & \omega^i_J \\ \omega^I_j & \omega^I_J \end{pmatrix}. $$ The structure equations of Cartan are $d\o …

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 …

The characteristic variety (i.e. vanishing locus of the symbol) of a symplectomorphism invariant scalar differential equation is a real projective hypersurface invariant under the group of projectiviz …

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 …