Abstract
We will be concerned with two open questions in symplectic geometry. The first question is the existence problem for lagrangian foliations. The second question is to know whether an affine manifold can be embedded as leaf of a lagrangian foliation. We prove that every homogeneous symplectic manifold with “closed regular” action of a solvable Lie group has lagrangian foliations. Moreover such manifold (M,ω) has a “bilagrangian linear connection (M,∇) such that ω is parallel with respect to ∇. About the second question, we prove that every connected and simply connected Lie group with left invariant affine structure can be embedded as leaf of a left invariant lagrangian foliation in a symplectic Lie group (G,ω).
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Nguiffo Boyom, M. Varietes symplectiques affines. Manuscripta Math 64, 1–33 (1989). https://doi.org/10.1007/BF01182083
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DOI: https://doi.org/10.1007/BF01182083