## Topological Methods in Nonlinear Analysis

### Poisson structures on closed manifolds

Sauvik Mukherjee

#### Abstract

We prove an $h$-principle for Poisson structures on closed manifolds. Equivalently, we prove an $h$-principle for symplectic foliations (singular) on closed manifolds. On open manifolds however the singularities could be avoided and it is a known result by Fernandes and Frejlich [Int. Math. Res. Not. IMRN (2012), no. 7, 1505-1518].

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 51, Number 1 (2018), 243-257.

Dates
First available in Project Euclid: 17 February 2018

https://projecteuclid.org/euclid.tmna/1518836437

Digital Object Identifier
doi:10.12775/TMNA.2017.059

Mathematical Reviews number (MathSciNet)
MR3784744

Zentralblatt MATH identifier
06887980

#### Citation

Mukherjee, Sauvik. Poisson structures on closed manifolds. Topol. Methods Nonlinear Anal. 51 (2018), no. 1, 243--257. doi:10.12775/TMNA.2017.059. https://projecteuclid.org/euclid.tmna/1518836437

#### References

• M. Bertelson, A $h$-principle for open relations invariant under foliated isotopies, J. Symplectic Geom. 1 (2002), 369–425.
• M. Bertelson, Foliations associated to regular Poisson structures, Commun. Contemp. Math. 3 (2001), no. 3, 441–456.
• M.S. Borman, Y. Eliashberg and E. Murphy, Existence and classification of overtwisted contact structures in all dimensions, Acta Math. 215 (2015), 215–281.
• J.P. Dufour and N.T. Zung, Poisson Structures and Their Normal Forms, Progress in Mathematics, Vol. 242, Birkhäuser, 2005.
• Y. Eliashberg and N.M. Mishachev, Wrinkling of smooth mappings and its applications I, Invent. Math. 130 (1997), 349–369.
• Y. Eliashberg and N.M. Mishachev, Introduction to the $h$-Principle, Graduate Studies in Mathematics, Vol. 48, American Mathematical Society, Providence, 2002.
• R.L. Fernandes and P. Frejlich, An $h$-principle for symplectic foliations, Int. Math. Res. Not. IMRN (2012), no. 7, 1505–1518.
• M. Gromov, Stable mappings of foliations into manifolds, Izv. Akad. Nauk SSSR Ser. Math. 33 (1069), 707–734 (in Russian).
• R. Thom, Les singularités des application différentiables, Ann. Inst. Fourier (Grenoble) 6 (1955/1956), 43–87.
• I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Progress in Mathematics, Vol. 118, Birkhäuser, 1994.