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Zbl 1035.53111
Vaisman, Izu
On locally Lagrangian symplectic structures. (English)
[A] Mladenov, Iva\"ilo M. (ed.) et al., Proceedings of the 4th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 6--15, 2002. Sofia: Coral Press Scientific Publishing. 326-329 (2003). ISBN 954-90618-4-1/pbk

Let $M$ be a manifold with local coordinates $x^1,\dots, x^n$ and let $\xi^1,\dots,\xi^n$ be the corresponding natural coordinates on the fibers of TM. The author introduces special symplectic forms of the kind $$\omega_L= \sum {\partial^2 L\over\partial x^i\partial\xi^j}\, dx^i\wedge dx^j+ \sum {\partial^2 L\over\partial\xi^i \partial\xi^j}\, d\xi^i\wedge dx^j,$$ where $L\in C^\infty(\text{TM})$ is a nondegenerate Lagrangian and moreover the tensor field $S\in \text{End\,TM}$ defined by $S(\partial/\partial x^i)= \partial/\partial\xi^i$, $S(\partial/\partial\xi^i)= 0$. Locally symplectic Lagrangian manifolds $M$ are such manifolds that are equipped with both objects $\omega_L$ and $S$. They are characterized without use of coordinates and more involved generalization in Poisson geometry is stated, too. No proofs are given.
[Jan Chrastina (Brno)]

MSC 2000:: *53D05 Symplectic manifolds, general

Keywords: symplectic manifold; locally Lagrangian symplectic structure; tangent manifold

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