# American Institute of Mathematical Sciences

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

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

 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  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.

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
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