Differential geometry
Classification of bm-Nambu structures of top degree
Comptes Rendus. Mathématique, Volume 356 (2018) no. 1, pp. 92-96.

In this paper, we classify bm-Nambu structures via bm-cohomology. The complex of bm-forms is an extension of the De Rham complex, which allows us to consider singular forms. bm-Cohomology is well understood thanks to Scott (2016) [12], and it can be expressed in terms of the De Rham cohomology of the manifold and of the critical hypersurface using a Mazzeo–Melrose-type formula. Each of the terms in bm-Mazzeo–Melrose formula acquires a geometrical interpretation in this classification. We also give equivariant versions of this classification scheme.

On classifie les structures bm-Nambu de degré maximal en utilisant la bm-cohomologie. Le complexe des bm-formes est une extension du complexe de De Rham et permet considérer des formes singulières. La bm-cohomologie est bien comprise grâce à Scott (2016) [12], et elle peut être exprimée en termes de la cohomologie de De Rham de la variété et de l'hypersurface critique en utilisant une formule de type Mazzeo–Melrose. Chacun des termes dans la formule de bm-Mazzeo–Melrose acquiert une interpretation géométrique dans cette classification. On donne aussi des versions équivariantes des théorèmes de classification.

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DOI: 10.1016/j.crma.2017.12.009
Miranda, Eva 1; Planas, Arnau 2

1 Laboratory of Geometry and Dynamical Systems, Department of Mathematics–UPC and BGSMath in Barcelona and CEREMADE (Université de Paris-Dauphine)– IMCCE (Observatoire de Paris)– IMJ (Université Paris-Diderot), Observatoire de Paris, 77, avenue Denfert-Rochereau, 75014 Paris, France
2 Laboratory of Geometry and Dynamical Systems, Department of Mathematics–UPC, Barcelona, Spain
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Miranda, Eva; Planas, Arnau. Classification of bm-Nambu structures of top degree. Comptes Rendus. Mathématique, Volume 356 (2018) no. 1, pp. 92-96. doi : 10.1016/j.crma.2017.12.009. http://www.numdam.org/articles/10.1016/j.crma.2017.12.009/

[1] Arnold, V.I. Remarks on Poisson structures on a plane and on other powers of volume elements, Tr. Semin. Petrovskogo, Volume 242 (1987) no. 12, pp. 37-46 (in Russian), translation in: J. Sov. Math., 47, 3, 1989, pp. 2509-2516

[2] Delshams, A.; Kiesenhofer, A.; Miranda, E. Examples of integrable and non-integrable systems on singular symplectic manifolds, J. Geom. Phys., Volume 115 (2017), pp. 89-97 | DOI

[3] Guillemin, V.; Miranda, E.; Pires, A.R. Symplectic and Poisson geometry on b-manifolds, Adv. Math., Volume 264 (2014), pp. 864-896

[4] Guillemin, V.; Miranda, E.; Pires, A.R.; Scott, G. Toric actions on b-symplectic manifolds, Int. Math. Res. Not., Volume 2015 (2015) no. 14, pp. 5818-5848 | DOI

[5] Guillemin, V.; Miranda, E.; Weitsman, J. Desingularizing bm-symplectic structures, Int. Math. Res. Not., Volume 2017 (2017) no. 6 | DOI

[6] Kiesenhofer, A.; Miranda, E. Cotangent models for integrable systems, Commun. Math. Phys., Volume 350 (2017) no. 3, pp. 1123-1145

[7] Marcut, I.; Osorno, B. Deformations of log-symplectic structures, J. Lond. Math. Soc. (2), Volume 90 (2014) no. 1, pp. 197-212

[8] Martinez-Torres, D. Global classification of generic multi-vector fields of top degree, J. Lond. Math. Soc., Volume 69 (2004) no. 3, pp. 751-766

[9] Melrose, R. Atiyah–Patodi–Singer Index Theorem, Res. Notes Math., A.K. Peters, Wellesley, MA, USA, 1993

[10] Moser, J. On the volume elements on a manifold, Trans. Amer. Math. Soc., Volume 120 (1965), pp. 286-294

[11] Nambu, Y. Generalized Hamiltonian dynamics, Phys. Rev. D, Volume 7 (1973) no. 8, pp. 2405-2412

[12] Scott, G. The geometry of bk manifolds, J. Symplectic Geom., Volume 14 (2016) no. 1, pp. 71-95

[13] Takhtajan, L. On foundation of the generalized Nambu mechanics, Commun. Math. Phys., Volume 160 (2014), pp. 295-315

Cited by Sources:

Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia 2016 Prize, by a Chaire d'excellence de la “Fondation Sciences mathématiques de Paris”, and is partially supported by the “Ministerio de Economía y Competitividad” project (reference MTM2015-69135-P/FEDER) and by the “Generalitat de Catalunya” project (reference 2014SGR634). This work is supported by a public grant overseen by the French National Research Agency (ANR) as part of the “Investissements d'avenir” program (reference ANR-10-LABX-0098). Arnau Planas is partially supported by the projects MTM2015-69135-P/FEDER and 2014SGR634.