On locally Lagrangian symplectic structures. (English) Zbl 1035.53111

Vaisman, Izu

Mladenov, Ivaïlo 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 (ISBN 954-90618-4-1/pbk). 326-329 (2003).

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.
For the entire collection see [Zbl 1008.00022].

Reviewer: Jan Chrastina (Brno)

MSC:

53D05

Symplectic manifolds (general theory)

Keywords:

symplectic manifold; locally Lagrangian symplectic structure; tangent manifold