Recherche

Livre

• P. CABAU, F. PELLETIER, Direct and Projective Limits of Geometric Banach Structures with the assistance and partial collaboration of Daniel Beltiţă. 1st Edition, Chapman & Hall/CRC Monographs and Research Notes in Mathematics (2023).

  Abstract: This book describes in detail the basic context of the Banach setting and the most important Lie structures found in finite dimension. The authors expose these concepts in the convenient framework which is a common context for projective and direct limits of Banach structures. The book presents sufficient conditions under which these structures exist by passing to such limits. In fact, such limits appear naturally in many mathematical and physical domains. Many examples in various fields illustrate the different concepts introduced.
  Many geometric structures, existing in the Banach setting, are "stable" by passing to projective and direct limits with adequate conditions. The convenient framework is used as a common context for such types of limits. The contents of this book can be considered as an introduction to differential geometry in infinite dimension but also a way for new research topics.
  This book allows the intended audience to understand the extension to the Banach framework of various topics in finite dimensional differential geometry and, moreover, the properties preserved by passing to projective and direct limits of such structures as a tool in different fields of research.

  

Publications

• P. CABAU, F. PELLETIER, Structures bihamiltoniennes partielles , arXiv:2207.11601v1 [math.DG] (2022).

  Résumé : Nous introduisons dans cet article trois notions de structures bihamiltoniennes partielles ($\operatorname{PQ}$, $\operatorname{PN}$ et $\operatorname{P\Omega}$) dans le cadre $c^\infty$-complet au sens de Frölicher, Kriegl et Michor (ou cadre adapté), étudions des objets géométriques liés à ces structures et en donnons des exemples à la fois en dimensions finie et infinie.
  Abstract: We introduce three notions of partial bihamiltonian structures ($\operatorname{PQ}$, $\operatorname{PN}$ et $\operatorname{P\Omega}$) in the convenient setting defined by Frölicher, Kriegl and Michor. We study geometrical objects linked with these structures and give examples in finite and infinite dimensions.

• P. CABAU, F. PELLETIER, Prolongations of convenient Lie algebroids, Mathematical Physics, Analysis and Geometry(2022) 25:17, arXiv:2007.10657 [math.DG].

  Abstract: We first define the concept of Lie algebroid in the convenient setting. In reference to the finite dimensional context, we adapt the notion of prolongation of a Lie algebroid over a fibred manifold to a convenient Lie algebroid over a fibred manifold. Then we show that this construction is stable under projective and direct limits under adequate assumptions.

• P. CABAU, F. PELLETIER, Projective and Direct limits of Banach $G$ and tensor structures, Differential Geometry - Dynamical Systems, Vol.22 (2020) 42—86 .

  Abstract: We endow projective (resp. direct) limits of Banach tensor structures with Fr\'{e}chet (resp. convenient) structures and study adapted connections to $G$-structures in both frameworks. This situation is illustrated by a lot of examples.

• P. CABAU, F. PELLETIER, Integrability on Direct Limits of Banach Manifolds, Annales de La Faculté des Sciences de Toulouse, Vol. 5 (2019) 909—956.
  [primary 58A30, 18A30, 46T05; secondary 17B66, 37K30, 22E65]

  Abstract: This paper is devoted to the framework of direct limit of anchored Banach bundles over a convenient manifold which is a direct limit of manifolds. In particular we give a criterion of integrability for distributions on such convenient manifolds which are locally direct limits of particular sequences of Banach anchor ranges.
  Résumé : Dans cet article, on s'intéresse à l'étude de divers objets rencontrés dans le cadre de limites directes de fibrés de Banach, munis d'une ancre, au dessus de certaines variétés apparaissant comme limites directes de variétés de Banach. En particulier, on donne un critère d'intégrabilité pour des distributions sur de telles variétés qui sont localement des limites directes de suites particulières d'images d'ancres banachiques.

• P. CABAU, Projective limits of local shift morphisms, Applied Sciences, Vol. 21 (2019) 69—83.
  [46A13, 46E20, 46G05, 46M40, 35R15, 35Q53]

  Abstract: We define the notion of projective limit of local shift morphisms of type $\left( r,s\right) $ and endow the space of such mathematical objects with an adapted differential structure. The notion of shift Poisson tensor $P$ on a Hilbert tower corresponds to such a morphism which is antisymmetric and whose Schouten bracket with itself $\left[ P,P\right] $ vanishes. We illustrate this notion with the example of the famous KdV equation on the circle $\mathbb{S}^{1}$ for which one can associate a pair of such compatible Poisson tensors on the Hilbert tower $\left( H^{n} \left( \mathbb{S}^{1} \right)\right) _{n\in\mathbb{N}^{\ast}}$.

• F. PELLETIER, P. CABAU, Convenient partial Poisson Manifolds, Journal of Geometry and Physics, Volume 136 (2019) 173--194.
  [58A30, 18A30, 46T05, 17B66, 37K30, 22E65]

  Abstract: We introduce the concept of partial Poisson structure on a manifold M modelled on a convenient space. This is done by specifying a (weak) subbundle $T^{\prime}M$ of $T^{\ast}M$ and an antisymmetric morphism $:T^{\prime}M \rightarrow TM$ such that the bracket defines a Poisson bracket on the algebra $\mathcal{A}$ of smooth functions $f$ on $M$ whose differential $df$ induces a section of $T^{\prime}M$. In particular, to each such function $f\in\mathcal{A}$ is associated a hamiltonian vector field $P(df)$. This notion takes naturally place in the framework of infinite dimensional weak symplectic manifolds and Lie algebroids. After having defined this concept, we will illustrate it by a lot of natural examples. We will also consider the particular situations of direct (resp. projective) limits of such Banach structures. Finally, we will also give some results on the existence of (weak) symplectic foliations naturally associated to some particular partial Poisson structures.

• P. CABAU, Límites directos de prolongaciones de algebroides de Lie, Revista de Matemática: Teoría y Aplicaciones, Vol. 24, Núm. 1 (2017) [en español].
  [46A13, 46T05, 58A32]

  Abstract: We prove that direct limits of finite dimensional Lie algebroids and their prolongations can be endowed with structures of convenient spaces.
  Resumen: Probamos que se puede definir estructuras de espacios convenientes sobre límites directos de algebroides de Lie y sus prolongaciones.
  Resumé : On établit que l’on peut munir les limites directes d’algébroïdes de Lie et de leurs prolongements de structures d’espaces $c^\infty$-complets (adapté).

• A. SURI, P. CABAU, Geometric structure for the tangent bundle of direct limit manifolds, Differential Geometry - Dynamical Systems, Vol.16 (2014) 239—247, arXiv:1308.3063v1 [math.DG]
  

  Abstract: We equip the direct limit of tangent bundles of paracompact finite dimensional manifolds with a structure of convenient vector bundle.

• P. CABAU, F. PELLETIER, Almost Lie structures on an anchored Banch bundle, Journal of Geometry and Physics 62 (2012) 2147—2169, arXiv:1111.5908v1 [math.DG].
  

  Abstract: Under appropriate assumptions, we generalize the concept of linear almost Poisson structures, almost Lie algebroids, almost differentials in the framework of Banach anchored bundles and the relation between these objects. We then obtain an adapted formalism for mechanical systems which is illustrated by the evolutionary problem of the "Hilbert snake".

• P. CABAU, Strong projective limits of Banach Lie algebroids, Portugaliae Mathematica, Volume 69, Issue 1 (2012).
  v.f. : Limites projectives fortes d'algébroïdes de Lie, arXiv:1012.0698v3 [math.DG].
  

  Abstract: We define the notion of strong projective limit of Banach Lie algebroids. We study the associated structures of Fréchet bundles and the compatibility with the different morphisms. This kind of structure seems to be a convenient framework for various situations.
  Résumé : On définit la notion de limite projective forte d'algébroïdes de Lie banachiques. On étudie les structures associées de fibrés fréchétiques et la compatibilité avec les divers morphismes. Ce type de structure semble être un cadre adapté pour des situations variées.

• P. CABAU, Generic Jacobi Manifolds, Differential Geometry – Dynamical Systems, Vol. 12 (2010) 41—51. [53B50, 57N80]
  

  Abstract: We build a stratification on the $1$-jets of pairs (vector fields, twice contravariant tensors) which allows to define the notion of finite-dimensional generic Jacobi manifold. We then describe the singularities of the associated characteristic field. We also give an example of such manifolds in thermodynamics.

• F. PELLETIER, P. CABAU, Generalized PN manifolds and Separation of Variables, Banach Center Publications 82 Geometry and Topology of Caustics (2008) 163—181. [37J05, 37J15, 37J20, 53B50, 53D05, 53D17, 53Z05, 53Z20]
  

  Abstract: The notion of generalized PN manifold is a framework which allows to get properties about first integrals of the associated bihamiltonian system: conditions of existence of a bi-abelian sub-algebra obtained from the momentum map and characterization of such an algebra linked with the problem ofseparation of variables.

• P. CABAU, Bilagrangian fanning curves, Far East Journal of Dynamical Systems, volume 10 (3) (2008) 293—305. [37J, 53D17, 53Z05]
  

  Abstract: We define the notion of bilagrangian fanning curves on $\mathbb{R}^{2n}$ endowed with a $\omega N$ structure; these curves are lagrangian and invariant with respect to the recursion operator $N.$We study some of their properties of and give an example of these curves in mechanics.

• A. BEN ABDESSELEM, P. CABAU, Special Lagrangian Manifolds obtained from Complex Grassmannians, International Journal of Pure and Applied Mathematics, Volume 30, n°3 (2006). [53C42, 53D12], arXiv:math.DG/0605389 v1, 15 May 2006

• P. CABAU, F. PELLETIER, Localisation des courbes anormales et couples de tenseurs de Poisson en petite dimension, Bulletin des Sciences Mathématiques 124 n°6 (2000) 459—515.  Prépublications du LAMA, Université de Savoie, 99-07c, 1999. [49J15, 53B50]
• P. CABAU, J. GRIFONE, M. MEHDI, Existence de lois de conservation dans le cas cyclique, Annales Institut Henri Poincaré, Physique théorique 55 n°3 (1991) 789—803. [58H10, 58F35, 58G35]
  Article réalisé au laboratoire de Topologie et Géométrie, Université Paul Sabatier, Toulouse III

Thèse

• Couples de tenseurs de Poisson compatibles et localisation de courbes anormales en petite dimension, Université de Savoie, 1999
  Résumé : On réalise une étude géométrique d'un couple de tenseurs de Poisson compatibles (dont le premier est de rang maximum) sur des variétés génériques de dimensions 4 et 5. Les stratifications utilisées pour obtenir de telles propriétés géométriques génériques sont réinvesties pour localiser des courbes anormales relatives à des distributions génériques de codimension 2 sur des variétés de dimensions 6 et 7.
  Abstract: We realize a geometrical study of a couple of compatible Poisson tensors (the first one is of maximal rank) on generic manifolds of dimensions 4 and 5. In order to get such generic geometric properties, we need to build stratifications which are also invested to localize abnormal curves for generic distributions of codimension 2 on manifolds of dimensions 6 and 7.

Encadrement de mémoires

• Contrôle optimal de systèmes sur des groupes de Lie, Mastère d'Ingénierie Mathématique (M2), 2009
• Contrôlabilité de systèmes affines invariants à gauche sur un groupe de Lie, Mastère d'Ingénierie Mathématique (M2), 2008