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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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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

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

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 $X_H$ …