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September  2020, 12(3): 343-361. doi: 10.3934/jgm.2020015

The method of averaging for Poisson connections on foliations and its applications

Misael Avendaño-Camacho 1,, , Isaac Hasse-Armengol 2, , Eduardo Velasco-Barreras 1,3, and  Yury Vorobiev 1,

1. 

Departamento de Matemáticas, Universidad de Sonora, Blvd. Luis Encinas y Rosales, s/n, Col. Centro, C.P. 83000, Hermosillo, Son., México
2. 

Departamento de Física y Matemáticas, Universidad Autónoma de Ciudad Juárez, Av. del Charro no. 450 nte., Col. Partido Romero, C.P. 32310, Ciudad Juárez, Chihuahua, México
3. 

Instituto de Matemática e Estatística, Universidade Federal Fluminense, Rua Professor Marcos Waldemar de Freitas Reis, s/n, Niterói 24210-201, Río de Janeiro, Brasil

* Corresponding author: Misael Avendaño-Camacho

Received  September 2019 Revised  February 2020 Published  September 2020 Early access  June 2020

Fund Project: The authors are supported by CONACYT grant CB2015 no. 258302. E. Velasco - Barreras was supported by FAPERJ grants E-26/202.411/2019 and E-26/202.412/2019

On a Poisson foliation equipped with a canonical and cotangential action of a compact Lie group, we describe the averaging method for Poisson connections. In this context, we generalize some previous results on Hannay-Berry connections for Hamiltonian and locally Hamiltonian actions on Poisson fiber bundles. Our main application of the averaging method for connections is the construction of invariant Dirac structures parametrized by the 2-cocycles of the de Rham-Casimir complex of the Poisson foliation.

Keywords: Poisson foliation, cotangential group action, invariant Ehresmann connections, averaging method, pre-momentum map, Dirac structure.
Mathematics Subject Classification: Primary: 53D17, 53C12; Secondary: 53D20, 58D19, 70G45.
Citation: Misael Avendaño-Camacho, Isaac Hasse-Armengol, Eduardo Velasco-Barreras, Yury Vorobiev. The method of averaging for Poisson connections on foliations and its applications. Journal of Geometric Mechanics, 2020, 12 (3) : 343-361. doi: 10.3934/jgm.2020015
References:
[1]

M. Avendaño-Camacho, J. A. Vallejo and Y. Vorobiev, Higher order corrections to adiabatic invariants of generalized slow-fast Hamiltonian systems, J. Math. Phys., 54 (2013), 15pp. doi: 10.1063/1.4817863.  Google Scholar

[2]

M. Avendaño-Camacho and Y. Vorobiev, Deformations of Poisson structures on fibered manifolds and adiabatic slow-fast systems, Int. J. Geom. Methods Mod. Phys., 14 (2017), 15pp. doi: 10.1142/S0219887817500864.  Google Scholar

[3]

O. Brahic and R. L. Fernandes, Integrability and reduction of Hamiltonian actions on Dirac manifolds, Indag. Math., 25 (2014), 901-925.  doi: 10.1016/j.indag.2014.07.007.  Google Scholar

[4]

O. Brahic and R. L. Fernandes, Poisson fibrations and fibered symplectic groupoids, in Poisson Geometry in Mathematics and Physics, Contemp. Math., 450, Amer. Math. Soc., Providence, RI, 2008, 41–59. doi: 10.1090/conm/450/08733.  Google Scholar

[5]

T. Courant and A. Weinstein, Beyond Poisson structures, in Action Hamiltoniennes de Groupes. Troisième Théorème de Lie, Travaux en Cours, 27, Hermann, Paris, 1988, 39–49.  Google Scholar

[6]

T. J. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631-661.  doi: 10.1090/S0002-9947-1990-0998124-1.  Google Scholar

[7]

J.-P. Dufour and A. Wade, On the local structure of Dirac manifolds, Compos. Math., 144 (2008), 774-786.  doi: 10.1112/S0010437X07003272.  Google Scholar

[8]

V. L. Ginzburg, Momentum mappings and Poisson cohomology, Internat. J. Math., 7 (1996), 329-358.  doi: 10.1142/S0129167X96000207.  Google Scholar

[9]

V. L. Ginzburg, Equivariant Poisson cohomology and a spectral sequence associated with a moment map, Internat. J. Math., 10 (1999), 977-1010.  doi: 10.1142/S0129167X99000422.  Google Scholar

[10]

I. Kolář, P. W. Michor and J. Slovák, Natural Operations in Differential Geometry, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-662-02950-3.  Google Scholar

[11]

J.-H. Lu, Momentum mappings and reduction of Poisson actions, in Symplectic Geometry, Groupoids, and Integrable Systems, Math. Sci. Res. Inst. Publ., 20, Springer, New York, NY, 1991,209–226. doi: 10.1007/978-1-4613-9719-9_15.  Google Scholar

[12]

J. Marsden, R. Montgomery and T. Ratiu, Reduction, symmetry, and phases in mechanics, Mem. Amer. Math. Soc., 88 (1990). doi: 10.1090/memo/0436.  Google Scholar

[13]

R. Montgomery, The connection whose holonomy is the classical adiabatic angles of Hannay and Berry and its generalization to the non-integrable case, Commun. Math. Phys., 120 (1988), 269-294.  doi: 10.1007/BF01217966.  Google Scholar

[14]

A. Pedroza, E. Velasco-Barreras and Y. Vorobiev, Unimodularity criteria for Poisson structures on foliated manifolds, Lett. Math. Phys., 108 (2018), 861-882.  doi: 10.1007/s11005-017-1014-3.  Google Scholar

[15]

M. R. Sepanski, Compact Lie Groups, Graduate Texts in Mathematics, 235, Springer, New York, 2007. doi: 10.1007/978-0-387-49158-5.  Google Scholar

[16]

P. Ševera and A. Weinstein, Poisson geometry with a 3-form background. Noncommutative geometry and string theory, Progr. Theoret. Phys. Suppl., 144 (2001), 145-154.  doi: 10.1143/PTPS.144.145.  Google Scholar

[17]

I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Progress in Mathematics, 118, Birkhäuser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8495-2.  Google Scholar

[18]

I. Vaisman, Coupling Poisson and Jacobi structures on foliated manifolds, Int. J. Geom. Methods Mod. Phys., 1 (2004), 607-637.  doi: 10.1142/S0219887804000307.  Google Scholar

[19]

I. Vaisman, Foliation-coupling Dirac structures, J. Geom. Phys., 56 (2006), 917-938.  doi: 10.1016/j.geomphys.2005.05.007.  Google Scholar

[20]

J. A. Vallejo and Y. Vorobiev, Invariant Poisson realizations and the averaging of Dirac structures, SIGMA Symmetry Integrability Geom. Methods Appl., 10 (2014), 20pp. doi: 10.3842/SIGMA.2014.096.  Google Scholar

[21]

Y. Vorobjev, Coupling tensors and Poisson geometry near a single symplectic leaf, in Lie Algebroids and Related Topics in Differential Geometry, Banach Center Publ., 54, Polish Acad. Sci. Inst. Math., Warsaw, 2001,249–274. doi: 10.4064/bc54-0-14.  Google Scholar

[22]

A. Wade, Poisson fiber bundles and coupling Dirac structures, Ann. Global Anal. Geom., 3 (2008), 207-217.  doi: 10.1007/s10455-007-9079-3.  Google Scholar

[23]

M. Wüstner, A connected Lie group equals the square of the exponential image, J. Lie Theory, 13 (2003), 307-309.   Google Scholar

show all references

References:
[1]

M. Avendaño-Camacho, J. A. Vallejo and Y. Vorobiev, Higher order corrections to adiabatic invariants of generalized slow-fast Hamiltonian systems, J. Math. Phys., 54 (2013), 15pp. doi: 10.1063/1.4817863.  Google Scholar

[2]

M. Avendaño-Camacho and Y. Vorobiev, Deformations of Poisson structures on fibered manifolds and adiabatic slow-fast systems, Int. J. Geom. Methods Mod. Phys., 14 (2017), 15pp. doi: 10.1142/S0219887817500864.  Google Scholar

[3]

O. Brahic and R. L. Fernandes, Integrability and reduction of Hamiltonian actions on Dirac manifolds, Indag. Math., 25 (2014), 901-925.  doi: 10.1016/j.indag.2014.07.007.  Google Scholar

[4]

O. Brahic and R. L. Fernandes, Poisson fibrations and fibered symplectic groupoids, in Poisson Geometry in Mathematics and Physics, Contemp. Math., 450, Amer. Math. Soc., Providence, RI, 2008, 41–59. doi: 10.1090/conm/450/08733.  Google Scholar

[5]

T. Courant and A. Weinstein, Beyond Poisson structures, in Action Hamiltoniennes de Groupes. Troisième Théorème de Lie, Travaux en Cours, 27, Hermann, Paris, 1988, 39–49.  Google Scholar

[6]

T. J. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631-661.  doi: 10.1090/S0002-9947-1990-0998124-1.  Google Scholar

[7]

J.-P. Dufour and A. Wade, On the local structure of Dirac manifolds, Compos. Math., 144 (2008), 774-786.  doi: 10.1112/S0010437X07003272.  Google Scholar

[8]

V. L. Ginzburg, Momentum mappings and Poisson cohomology, Internat. J. Math., 7 (1996), 329-358.  doi: 10.1142/S0129167X96000207.  Google Scholar

[9]

V. L. Ginzburg, Equivariant Poisson cohomology and a spectral sequence associated with a moment map, Internat. J. Math., 10 (1999), 977-1010.  doi: 10.1142/S0129167X99000422.  Google Scholar

[10]

I. Kolář, P. W. Michor and J. Slovák, Natural Operations in Differential Geometry, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-662-02950-3.  Google Scholar

[11]

J.-H. Lu, Momentum mappings and reduction of Poisson actions, in Symplectic Geometry, Groupoids, and Integrable Systems, Math. Sci. Res. Inst. Publ., 20, Springer, New York, NY, 1991,209–226. doi: 10.1007/978-1-4613-9719-9_15.  Google Scholar

[12]

J. Marsden, R. Montgomery and T. Ratiu, Reduction, symmetry, and phases in mechanics, Mem. Amer. Math. Soc., 88 (1990). doi: 10.1090/memo/0436.  Google Scholar

[13]

R. Montgomery, The connection whose holonomy is the classical adiabatic angles of Hannay and Berry and its generalization to the non-integrable case, Commun. Math. Phys., 120 (1988), 269-294.  doi: 10.1007/BF01217966.  Google Scholar

[14]

A. Pedroza, E. Velasco-Barreras and Y. Vorobiev, Unimodularity criteria for Poisson structures on foliated manifolds, Lett. Math. Phys., 108 (2018), 861-882.  doi: 10.1007/s11005-017-1014-3.  Google Scholar

[15]

M. R. Sepanski, Compact Lie Groups, Graduate Texts in Mathematics, 235, Springer, New York, 2007. doi: 10.1007/978-0-387-49158-5.  Google Scholar

[16]

P. Ševera and A. Weinstein, Poisson geometry with a 3-form background. Noncommutative geometry and string theory, Progr. Theoret. Phys. Suppl., 144 (2001), 145-154.  doi: 10.1143/PTPS.144.145.  Google Scholar

[17]

I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Progress in Mathematics, 118, Birkhäuser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8495-2.  Google Scholar

[18]

I. Vaisman, Coupling Poisson and Jacobi structures on foliated manifolds, Int. J. Geom. Methods Mod. Phys., 1 (2004), 607-637.  doi: 10.1142/S0219887804000307.  Google Scholar

[19]

I. Vaisman, Foliation-coupling Dirac structures, J. Geom. Phys., 56 (2006), 917-938.  doi: 10.1016/j.geomphys.2005.05.007.  Google Scholar

[20]

J. A. Vallejo and Y. Vorobiev, Invariant Poisson realizations and the averaging of Dirac structures, SIGMA Symmetry Integrability Geom. Methods Appl., 10 (2014), 20pp. doi: 10.3842/SIGMA.2014.096.  Google Scholar

[21]

Y. Vorobjev, Coupling tensors and Poisson geometry near a single symplectic leaf, in Lie Algebroids and Related Topics in Differential Geometry, Banach Center Publ., 54, Polish Acad. Sci. Inst. Math., Warsaw, 2001,249–274. doi: 10.4064/bc54-0-14.  Google Scholar

[22]

A. Wade, Poisson fiber bundles and coupling Dirac structures, Ann. Global Anal. Geom., 3 (2008), 207-217.  doi: 10.1007/s10455-007-9079-3.  Google Scholar

[23]

M. Wüstner, A connected Lie group equals the square of the exponential image, J. Lie Theory, 13 (2003), 307-309.   Google Scholar

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