Related papers: A property of conformally Hamiltonian vector field…

Let $(M, \Omega)$ be a holomorphically symplectic manifold equipped with a holomorphic Lagrangian fibration $\pi: M \to B$, and $\eta$ a closed $(1,1)$-form on $B$. Then $\Omega+ \pi^* \eta$ is a holomorphically symplectic form on a complex…

This paper defines a symplectic form on the infinite dimensional Fr\'echet manifold of framed curves of fixed length over a simply connected Riemannian manifold of constant curvature. The paper then considers Hamiltonian systems generated…

We show that there exist infinite-dimensional quasi-flats in the compactly supported Hamiltonian diffeomorphism group of the Liouville domain, with respect to the spectral norm, if and only if the symplectic cohomology of this Liouville…

We exhibit many examples of closed symplectic manifolds on which there is an autonomous Hamiltonian whose associated flow has no nonconstant periodic orbits (the only previous explicit example in the literature was the torus T^2n (n\geq 2)…

We use quantum and Floer homology to construct (partial) quasi-morphisms on the universal cover of the group of compactly supported Hamiltonian diffeomorphisms for a certain class of non-closed strongly semi-positive symplectic manifolds…

Let $M$ be a non-compact connected manifold with a cocompact and properly discontinuous action of a discrete group $G$. We establish a Poincar\'{e}-Hopf theorem for a bounded vector field on $M$ satisfying a mild condition on zeros. As an…

Given a fibration of a symplectic manifold by lagrangian tori, we show that each symplectic vector field splits into two parts : the first is Hamiltonian and the second is symplectic and preserves the fibration. We then show an application…

Let $(M,g)$ be a compact K\"ahler manifold and $f$ a positive smooth function such that its Hamiltonian vector field $K = J\mathrm{grad}_g f$ for the K\"ahler form $\omega_g$ is a holomorphic Killing vector field. We say that the pair…

In this paper, we present algebraic tools to obtain normal forms of $\omega$-Hamiltonian vector fields under a semisymplectic action of a Lie group, by taking into account the symmetries and reversing symmetries of the vector field. The…

When $G$ is a complex reductive algebraic group and $G/K$ is a reductive symmetric space, the decomposition of $\C[G/K]$ as a $K$-module was obtained (in a non-constructive way) by Richardson, generalizing the celebrated result of…

The main result is the identification of the orthogonal complement of the subalgebra of conformal vector field inside the algebra of all vector fields of a compact flat 2-manifold. As a fundamental tool, the complete Hodge decomposition for…

The `Generalized Symmetric Teleparallel Gravity' (GSTG) does not admit diffeomorphic invariance, since the auxiliary field as well as the shift vector act as non-propagating dynamical variables carrying 1/2 degrees of freedom each. We show…

Symplectic integration algorithms are well-suited for long-term integrations of Hamiltonian systems because they preserve the geometric structure of the Hamiltonian flow. However, this desirable property is generally lost when adaptive…

Let $(S,\omega)$ be a closed connected oriented surface whose genus $l$ is at least two equipped with a symplectic form. Then we show the vanishing of the cup product of the fluxes of commuting symplectomorphisms. This result may be…

Given a triangulated region in the complex plane, a discrete vector field $Y$ assigns a vector $Y_i\in \mathbb{C}$ to every vertex. We call such a vector field holomorphic if it defines an infinitesimal deformation of the triangulation that…

The symplectic cohomology of certain symplectic manifolds $W$ with non-compact ends modelled on the positive symplectization of a compact contact manifold $Y$ is shown to vanish whenever there is a positive loop of contactomorphisms of $Y$…

We show, by an elementary and explicit construction, that the group of Hamiltonian diffeomorphisms of certain symplectic manifolds, endowed with Hofer's metric, contains subgroups quasi-isometric to Euclidean spaces of arbitrary dimension.

This series of papers is devoted to an open-ended project aimed at the solution of Hilbert's sixth problem (concerning joint axiomatization of physics and probability theory) proposed to be constructed in the framework of an all-embracing…

We exhibit two three-parameter families of locally conformal symplectic forms on the solvmanifold $M_{n,k}$ considered in [1], and show, using the Hodge-de Rham theory for the Lichnerowicz cohomology that that they are not $d_{\omega}$…

The purpose of this article is to characterize symplectic and Hamiltonian circle actions on symplectic manifolds in terms of symplectic embeddings of Riemann surfaces. More precisely, we will show that (1) if $(M,\omega)$ admits a…