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On locally Lagrangian symplectic structures. (English) Zbl 1035.53111
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 ,,x n and let ξ 1 ,,ξ n be the corresponding natural coordinates on the fibers of TM. The author introduces special symplectic forms of the kind

ω L = 2 L x i ξ j dx i dx j + 2 L ξ i ξ j dξ i dx j ,

where LC (TM) is a nondegenerate Lagrangian and moreover the tensor field SEndTM defined by S(/x i )=/ξ i , S(/ξ i )=0. Locally symplectic Lagrangian manifolds M are such manifolds that are equipped with both objects ω L and S. They are characterized without use of coordinates and more involved generalization in Poisson geometry is stated, too. No proofs are given.

MSC:
53D05Symplectic manifolds, general