References & Citations
Mathematics > Dynamical Systems
Title: Infinite Lifting of an Action of Symplectomorphism Group on the set of Bi-Lagrangian Structures
(Submitted on 11 Jul 2021 (v1), last revised 2 Jun 2022 (this version, v5))
Abstract: We consider a smooth $2n$-manifold $M$ endowed with a bi-Lagrangian structure $(\omega,\mathcal{F}_{1},\mathcal{F}_{2})$. That is, $\omega$ is a symplectic form and $(\mathcal{F}_{1},\mathcal{F}_{2})$ is a pair of transversal Lagrangian foliations on $(M, \omega)$. Such structures have an important geometric object called the Hess Connection. Among the many importance of these connections, they allow to classify affine bi-Lagrangian structures.
In this work, we show that a bi-Lagrangian structure on $M$ can be lifted as a bi-Lagrangian structure on its trivial bundle $M\times\mathbb{R}^n$. Moreover, the lifting of an affine bi-Lagrangian structure is also an affine bi-Lagrangian structure. We define a dynamic on the symplectomorphism group and the set of bi-Lagrangian structures (that is an action of the symplectomorphism group on the set of bi-Lagrangian structures). This dynamic is compatible with Hess connections, preserves affine bi-Lagrangian structures, and can be lifted on $M\times\mathbb{R}^n$. This lifting can be lifted again on $\left(M\times\mathbb{R}^{2n}\right)\times\mathbb{R}^{4n}$, and coincides with the initial dynamic (in our sense) on $M\times\mathbb{R}^n$ for some bi-Lagrangian structures. Results still hold by replacing $M\times\mathbb{R}^{2n}$ with the tangent bundle $TM$ of $M$ or its cotangent bundle $T^{*}M$ for some manifolds $M$.
Submission history
From: Bertuel Tangue Ndawa [view email][v1] Sun, 11 Jul 2021 23:48:14 GMT (101kb,D)
[v2] Mon, 2 Aug 2021 15:29:24 GMT (101kb,D)
[v3] Wed, 2 Mar 2022 16:21:34 GMT (18kb,D)
[v4] Wed, 16 Mar 2022 08:50:33 GMT (19kb,D)
[v5] Thu, 2 Jun 2022 11:48:18 GMT (18kb,D)
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