Classification of bm-Nambu structures of top degree

Eva Miranda; Arnau Planas

doi:10.1016/j.crma.2017.12.009

Differential geometry

Classification of b^m-Nambu structures of top degree

Presented by: the Editorial Board

Eva Miranda ¹; Arnau Planas 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

Comptes Rendus. Mathématique, Volume 356 (2018) no. 1, pp. 92-96.

Abstract
Résumé

In this paper, we classify $b^{m}$ -Nambu structures via $b^{m}$ -cohomology. The complex of $b^{m}$ -forms is an extension of the De Rham complex, which allows us to consider singular forms. $b^{m}$ -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 $b^{m}$ -Mazzeo–Melrose formula acquires a geometrical interpretation in this classification. We also give equivariant versions of this classification scheme.

On classifie les structures $b^{m}$ -Nambu de degré maximal en utilisant la $b^{m}$ -cohomologie. Le complexe des $b^{m}$ -formes est une extension du complexe de De Rham et permet considérer des formes singulières. La $b^{m}$ -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 $b^{m}$ -Mazzeo–Melrose acquiert une interpretation géométrique dans cette classification. On donne aussi des versions équivariantes des théorèmes de classification.

Received: 2017-12-06
Accepted: 2017-12-19
Published online: 2017-12-28

DOI: 10.1016/j.crma.2017.12.009

Author's affiliations:

Eva Miranda ¹; Arnau Planas 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

@article{CRMATH_2018__356_1_92_0,
     author = {Eva Miranda and Arnau Planas},
     title = {Classification of {\protect\emph{b}\protect\textsuperscript{\protect\emph{m}}-Nambu} structures of top degree},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {92--96},
     publisher = {Elsevier},
     volume = {356},
     number = {1},
     year = {2018},
     doi = {10.1016/j.crma.2017.12.009},
     language = {en},
}

TY  - JOUR
AU  - Eva Miranda
AU  - Arnau Planas
TI  - Classification of bm-Nambu structures of top degree
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 92
EP  - 96
VL  - 356
IS  - 1
PB  - Elsevier
DO  - 10.1016/j.crma.2017.12.009
LA  - en
ID  - CRMATH_2018__356_1_92_0
ER  -

%0 Journal Article
%A Eva Miranda
%A Arnau Planas
%T Classification of bm-Nambu structures of top degree
%J Comptes Rendus. Mathématique
%D 2018
%P 92-96
%V 356
%N 1
%I Elsevier
%R 10.1016/j.crma.2017.12.009
%G en
%F CRMATH_2018__356_1_92_0

Eva Miranda; Arnau Planas. Classification of b^m-Nambu structures of top degree. Comptes Rendus. Mathématique, Volume 356 (2018) no. 1, pp. 92-96. doi : 10.1016/j.crma.2017.12.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.12.009/

References
Cited by

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

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

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

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

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

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

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

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

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

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

[12] G. Scott The geometry of $b^{k}$ manifolds, J. Symplectic Geom., Volume 14 (2016) no. 1, pp. 71-95

[13] L. Takhtajan 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.

Comments - Policy