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Some properties of tangent Dirac structures of higher order

P. M. Kouotchop Wamba, A. Ntyam, J. Wouafo Kamga (2012)

Archivum Mathematicum

Let M be a smooth manifold. The tangent lift of Dirac structure on M was originally studied by T. Courant in [3]. The tangent lift of higher order of Dirac structure L on M has been studied in [10], where tangent Dirac structure of higher order are described locally. In this paper we give an intrinsic construction of tangent Dirac structure of higher order denoted by L r and we study some properties of this Dirac structure. In particular, we study the Lie algebroid and the presymplectic foliation...

Submanifold averaging in Riemannian and symplectic geometry

Marco Zambon (2006)

Journal of the European Mathematical Society

We give a canonical construction of an “isotropic average” of given C 1 -close isotropic submanifolds of a symplectic manifold. For this purpose we use an improvement (obtained in collaboration with H. Karcher) of Weinstein’s submanifold averaging theorem and apply “Moser’s trick”. We also present an application to Hamiltonian group actions.

Sur les propriétés topologiques des projections lagrangiennes en géométrie symplectique des caustiques.

V. I. Arnold (1995)

Revista Matemática de la Universidad Complutense de Madrid

La caustique d?un point sur une variété riemannienne est l?ensemble des points d?intersection des géodésiques infiniment voisins partant de ce point. Jacobi a remarqué, en utilisant un raisonnement topologique, que la caustique d?un point sur une surface convexe fermée doit avoir des points de rebroussement. Il a aussi annoncé (sans démonstration) que le nombre de ces points est quatre pour les caustiques sur les surfaces d?ellipsoïdes (Jacobi, 1964). Dans cette note j?essaie d?inclure les théorèmes...

Symplectic Capacities in Manifolds

Alfred Künzle (1997)

Banach Center Publications

Symplectic capacities coinciding on convex sets in the standard symplectic vector space are extended to any subsets of symplectic manifolds. It is shown that, using embeddings of non-smooth convex sets and a product formula, calculations of some capacities become very simple. Moreover, it is proved that there exist such capacities which are distinct and that there are star-shaped domains diffeomorphic to the ball but not symplectomorphic to any convex set.

Symplectic connections with parallel Ricci tensor

Michel Cahen, Simone Gutt, John Rawnsley (2000)

Banach Center Publications

A variational principle introduced to select some symplectic connections leads to field equations which, in the case of the Levi Civita connection of Kähler manifolds, are equivalent to the condition that the Ricci tensor is parallel. This condition, which is stronger than the field equations, is studied in a purely symplectic framework.

Symplectic Killing spinors

Svatopluk Krýsl (2012)

Commentationes Mathematicae Universitatis Carolinae

Let ( M , ω ) be a symplectic manifold admitting a metaplectic structure (a symplectic analogue of the Riemannian spin structure) and a torsion-free symplectic connection ∇ . Symplectic Killing spinor fields for this structure are sections of the symplectic spinor bundle satisfying a certain first order partial differential equation and they are the main object of this paper. We derive a necessary condition which has to be satisfied by a symplectic Killing spinor field. Using this condition one may easily...

Symplectic spinor valued forms and invariant operators acting between them

Svatopluk Krýsl (2006)

Archivum Mathematicum

Exterior differential forms with values in the (Kostant’s) symplectic spinor bundle on a manifold with a given metaplectic structure are decomposed into invariant subspaces. Projections to these invariant subspaces of a covariant derivative associated to a torsion-free symplectic connection are described.

Symplectic twistor operator and its solution space on ℝ 2

Marie Dostálová, Petr Somberg (2013)

Archivum Mathematicum

We introduce the symplectic twistor operator T s in symplectic spin geometry of real dimension two, as a symplectic analogue of the Dolbeault operator in complex spin geometry of complex dimension 1. Based on the techniques of the metaplectic Howe duality and algebraic Weyl algebra, we compute the space of its solutions on ℝ 2 .

Systèmes hamiltoniens k-symplectiques.

Azzouz Awane, Mohamed Belam, Sadik Fikri, Mohammed Lahmouz, Bouchra Naanani (2002)

Revista Matemática Complutense

We study some properties of the k-symplectic Hamiltonian systems in analogy with the well-known classical Hamiltonian systems. The integrability of k-symplectic Hamiltonian systems and the relationships with the Nambu's statistical mechanics are given.

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