Hamiltonian vector fields on almost symplectic manifolds

Abstract

Let $(M,\omega )$ be an almost symplectic manifold ($\omega$ is a non degenerate, not closed, 2-form). We say that a vector field $X$ of $M$ is locally Hamiltonian if $L_X\omega =0,d(i(X)\omega )=0$, and it is Hamiltonian if, furthermore, the 1-form $i(X)\omega$ is exact. Such vector fields were considered in a 2007 paper by F. Fasso and N. Sansonetto, under the name of strongly Hamiltonian, and a corresponding action-angle theorem was proven. Almost symplectic manifolds may have few, non-zero, Hamiltonian vector fields or even none. Therefore, it is important to have examples and it is our aim to provide such examples here. We also obtain some new general results. In particular, we show that the locally Hamiltonian vector fields generate a Dirac structure on $M$ and we state a reduction theorem of the Marsden-Weinstein type. A final section is dedicated to almost symplectic structures on tangent bundles.

LaTex, 18 pages

References in corpus (1)

• Towards a double field theory on para-Hermitian manifolds

Cited by in corpus (7)

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• The super-Sasaki metric on the antitangent bundle
• Homogeneous bi-Hamiltonian structures and integrable contact systems
• Reduction of hybrid Hamiltonian systems with non-equivariant momentum maps