This paper is devoted to the framework of direct limit of anchored Banach bundles over a convenie... more This paper is devoted to the framework of direct limit of anchored Banach bundles over a convenient manifold which is a direct limit of Banach manifold. 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.
We first define the concept of Lie algebroid in the convenient setting. In reference to the finit... more 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.
We endow projective (resp. direct) limits of Banach tensor structures with Fr\'{e}chet (resp.... more 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.
We define the notion of projective limit of local shift morphisms of type $\left( r,s\right) $ an... more 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 morphisms which are antisymmetric and whose Schouten bracket $\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 couple of compatible Poisson tensors of this type on the Hilbert tower $\left( H^{n}(\mathbb{S}^{1})\right) _{n\in\mathbb{N}% ^{\ast}}$.
We equip the direct limit of tangent bundles of paracompact finite dimensional manifolds with a s... more We equip the direct limit of tangent bundles of paracompact finite dimensional manifolds with a structure of convenient vector bundle with structural group GL(∞,R) = lim→ GL(R n ).
We define the notion of strong projective limit of Banach Lie algebroids. We study the associated... more We define the notion of strong projective limit of Banach Lie algebroids. We study the associated structures of Fr\'{e}chet bundles and the compatibility with the different morphisms. This kind of structure seems to be a convenient framework for various situations.
We endow projective (resp. direct) limits of BanachG-stuctures and tensor structures with Fréchet... more We endow projective (resp. direct) limits of BanachG-stuctures and tensor structures with Fréchet (resp. convenient) structures. These situations are illustrated by many examples in the framework of tensor structures of type (1, 1) and (2, 0). M.S.C. 2010: 53C10, 58B25, 18A30, 22E65.
We define the notion of projective limit of local shift morphisms of type (r, s) and endow the sp... more We define the notion of projective limit of local shift morphisms of type (r, s) 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 [P, P ] vanishes. We illustrate this notion with the example of the famous KdV equation on the circle S for which one can associate a pair of such compatible Poisson tensors on the Hilbert tower ( H(S) ) n∈N∗ . M.S.C. 2010: 46A13, 46E20, 46G05, 46M40, 35R15, 35Q53.
We introduce the concept of partial Poisson structure on a manifold M modelled on a convenient sp... more 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 ′ M of T * M and an antisymmetric morphism P : T ′ M → T M such that the bracket {f, g} P = − < df, P (dg) > defines a Poisson bracket on the algebra A of smooth functions f on M whose differential df induces a section of T ′ M. In particular, to each such function f ∈ 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.
Probamos que se puede definir estructuras de espacios convenientes sobre límites directos de alge... more Probamos que se puede definir estructuras de espacios convenientes sobre límites directos de algebroides de Lie y sus prolongaciones.
This paper gives an example of special Lagrangian manifold obtained from a hypersurface of a comp... more This paper gives an example of special Lagrangian manifold obtained from a hypersurface of a complex Grassmannian with vanishing first Chern class. The obtained manifold is a 1-torus bundle over the two dimensional real projective space. Such manifolds are interesting for mirror symmetry theory. Other examples of the same type are provided at the end of this article. Introduction. In 1982, when F. Harvey and H. Lawson introduced Special Lagrangian manifolds in [7], their main interest was calibration problems. Now, these manifolds give another approach to mirror symmetry and string theory and become crucial in these fields. A conjecture due to Strominger, Yau and Zaslow in [9] explains mirror symmetry in a fairly
We define the notion of strong projective limit of Banach Lie algebroids. We study the associated... more We define the notion of strong projective limit of Banach Lie algebroids. We study the associated structures of Fr\'{e}chet bundles and the compatibility with the different morphisms. This kind of structure seems to be a convenient framework for various situations.
This paper is devoted to the framework of direct limit of anchored Banach bundles over a convenie... more This paper is devoted to the framework of direct limit of anchored Banach bundles over a convenient manifold which is a direct limit of Banach manifold. 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.
We equip the direct limit of tangent bundles of paracompact finite dimensional manifolds with a s... more We equip the direct limit of tangent bundles of paracompact finite dimensional manifolds with a structure of convenient vector bundle with structural group $GL(\infty,\mathbb{R}) =\underrightarrow{\lim}GL(\mathbb{R}^{n})$.
We prove that direct limits of finite dimensional Lie algebroids and their prolongations can be e... more We prove that direct limits of finite dimensional Lie algebroids and their prolongations can be endowed with structures of convenient spaces.
We build a stratification on the 1-jets of pairs (vector fields, twice contravariant tensors) whi... more 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.
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Papers by Patrick CABAU