arXiv subjects
Publications and source records attributed to P. Cabau.
We endow projective (resp. direct) limits of Banach tensor structures with Fréchet (resp. convenient) structures and study adapted connections to $G$-structures in both frameworks. This situation is illustrated by a lot of examples.
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 $P:T^{\prime}M\rightarrow TM$ such that the bracket $\{f,g\}_{P}=- $ 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.
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.