doi: 10.3934/jgm.2022006
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Infinite lifting of an action of symplectomorphism group on the set of bi-Lagrangian structures

University Institute of Technology, of the University of Ngaoundere, Cameroon

To Kira and Thierry Rothen’s family.

Received  December 2021 Revised  March 2022 Early access April 2022

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 a structure has an important geometric object called the Hess Connection. Among the many importance of Hess 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 affine. 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 $. By replacing $ M\times\mathbb{R}^{2n} $ with the tangent bundle $ TM $ or cotangent bundle $ T^{*}M $ of $ M $, results still hold when $ M $ is parallelizable.

Citation: Bertuel Tangue Ndawa. Infinite lifting of an action of symplectomorphism group on the set of bi-Lagrangian structures. Journal of Geometric Mechanics, doi: 10.3934/jgm.2022006
References:
[1]

M. N. Boyom, Variétés symplectiques affine, Manuscripta Math., 64 (1989), 1-33.  doi: 10.1007/BF01182083.

[2]

M. N. Boyom, The cohomology of Koszul-Vinberg algebras, Pacific J. Math., 225 (2006), 119-153.  doi: 10.2140/pjm.2006.225.119.

[3]

N. N. Boyom, Métriques kählériennes affinement plates de certaines variétés symplectiques, Proc. London. Math. Soc., 3-66 (1993), 338-380.  doi: 10.1112/plms/s3-66.2.358.

[4]

N. N. Boyom, Structures localement plates de certaines variétés symplectiques, Math. Scand., 76 (1995), 61-84.  doi: 10.7146/math.scand.a-12525.

[5]

A. C. da Silva, Lectures on Symplectic Geometry, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-540-45330-7.

[6]

F. Etayo and R. Santamaria, The canonical connection of a bi-Lagrangian manifold, J. Phys. A. Math. Gen., 34 (2001), 981-987.  doi: 10.1088/0305-4470/34/5/304.

[7]

F. Etayo and R. Santamaria, Bi-Lagrangian manifolds and associated geometric structures, Proceedings of the X Fall Workshop on Geometry and Physics, Miraflores de la Sierra, Madrid, Spain, (2003), 117–126.

[8]

F. EtayoR. Santamaria and U. R. Trías, The geometry of a bi-Lagrangian manifold, Differ. Geom. Appl., 24 (2006), 33-59.  doi: 10.1016/j.difgeo.2005.07.002.

[9]

H. Hess, Connections on symplectic manifolds and geometric quantization, Lecture Notes in Math., 836 (1980), 153-166. 

[10]

H. B. Lawson, Foliations, Bull. Am. Math. Soc., 80 (1974), 369-418.  doi: 10.1090/S0002-9904-1974-13432-4.

[11]

P. Libermann and C. M. Marle, Symplectic Geometry and Analytical Mechanics, D. Reidel Publishing Company, Dordrecht, Holland, 1987. doi: 10.1007/978-94-009-3807-6.

[12]

B. Loustau and A. Sanders, Bi-Lagrangian structures and Teichmüller theory, preprint, 2017, arXiv: 1708.09145.

[13]

E. Miranda and F. Presas, Geometric quantization of real polarization via sheaves, J. Symplectic Geom., 13 (2015), 421-462.  doi: 10.4310/JSG.2015.v13.n2.a6.

[14]

I. Vaisman, Hessian geometry on lagrange spaces, Int. J. Math. Math. Sci., 2014 (2014), Art. ID 793473, 9 pp. doi: 10.1155/2014/793473.

[15]

A. Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Advances in Math., 6 (1971), 329-346.  doi: 10.1016/0001-8708(71)90020-X.

show all references

References:
[1]

M. N. Boyom, Variétés symplectiques affine, Manuscripta Math., 64 (1989), 1-33.  doi: 10.1007/BF01182083.

[2]

M. N. Boyom, The cohomology of Koszul-Vinberg algebras, Pacific J. Math., 225 (2006), 119-153.  doi: 10.2140/pjm.2006.225.119.

[3]

N. N. Boyom, Métriques kählériennes affinement plates de certaines variétés symplectiques, Proc. London. Math. Soc., 3-66 (1993), 338-380.  doi: 10.1112/plms/s3-66.2.358.

[4]

N. N. Boyom, Structures localement plates de certaines variétés symplectiques, Math. Scand., 76 (1995), 61-84.  doi: 10.7146/math.scand.a-12525.

[5]

A. C. da Silva, Lectures on Symplectic Geometry, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-540-45330-7.

[6]

F. Etayo and R. Santamaria, The canonical connection of a bi-Lagrangian manifold, J. Phys. A. Math. Gen., 34 (2001), 981-987.  doi: 10.1088/0305-4470/34/5/304.

[7]

F. Etayo and R. Santamaria, Bi-Lagrangian manifolds and associated geometric structures, Proceedings of the X Fall Workshop on Geometry and Physics, Miraflores de la Sierra, Madrid, Spain, (2003), 117–126.

[8]

F. EtayoR. Santamaria and U. R. Trías, The geometry of a bi-Lagrangian manifold, Differ. Geom. Appl., 24 (2006), 33-59.  doi: 10.1016/j.difgeo.2005.07.002.

[9]

H. Hess, Connections on symplectic manifolds and geometric quantization, Lecture Notes in Math., 836 (1980), 153-166. 

[10]

H. B. Lawson, Foliations, Bull. Am. Math. Soc., 80 (1974), 369-418.  doi: 10.1090/S0002-9904-1974-13432-4.

[11]

P. Libermann and C. M. Marle, Symplectic Geometry and Analytical Mechanics, D. Reidel Publishing Company, Dordrecht, Holland, 1987. doi: 10.1007/978-94-009-3807-6.

[12]

B. Loustau and A. Sanders, Bi-Lagrangian structures and Teichmüller theory, preprint, 2017, arXiv: 1708.09145.

[13]

E. Miranda and F. Presas, Geometric quantization of real polarization via sheaves, J. Symplectic Geom., 13 (2015), 421-462.  doi: 10.4310/JSG.2015.v13.n2.a6.

[14]

I. Vaisman, Hessian geometry on lagrange spaces, Int. J. Math. Math. Sci., 2014 (2014), Art. ID 793473, 9 pp. doi: 10.1155/2014/793473.

[15]

A. Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Advances in Math., 6 (1971), 329-346.  doi: 10.1016/0001-8708(71)90020-X.

Figure 1.  The bi-Lagrangian structure $ (\mathcal{F}^x,\mathcal{F}^y) $
Figure 2.  The bi-Lagrangian structure $ (\mathcal{P},\mathcal{F}^y) $
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