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Infinite-dimensional hyperkähler manifolds associated with Hermitian-symmetric affine coadjoint orbits

Alice Barbara Tumpach (2009)

Annales de l’institut Fourier

In this paper, we construct a hyperkähler structure on the complexification 𝒪 ℂ of any Hermitian symmetric affine coadjoint orbit 𝒪 of a semi-simple L * -group of compact type, which is compatible with the complex symplectic form of Kirillov-Kostant-Souriau and restricts to the Kähler structure of 𝒪 . By a relevant identification of the complex orbit 𝒪 ℂ with the cotangent space T 𝒪 of 𝒪 induced by Mostow’s decomposition theorem, this leads to the existence of a hyperkähler structure on T 𝒪 compatible with...

Infinitesimal automorphisms and deformations of parabolic geometries

Andreas Čap (2008)

Journal of the European Mathematical Society

We show that infinitesimal automorphisms and infinitesimal deformations of parabolic geometries can be nicely described in terms of the twisted de Rham sequence associated to a certain linear connection on the adjoint tractor bundle. For regular normal geometries, this description can be related to the underlying geometric structure using the machinery of BGG sequences. In the locally flat case, this leads to a deformation complex, which generalizes the well known complex for locally conformally...

Infinitesimal conjugacies and Weil-Petersson metric

Albert Fathi, L. Flaminio (1993)

Annales de l'institut Fourier

We study deformations of compact Riemannian manifolds of negative curvature. We give an equation for the infinitesimal conjugacy between geodesic flows. This in turn allows us to compute derivatives of intersection of metrics. As a consequence we obtain a proof of a theorem of Wolpert.

Inner and outer hamiltonian capacities

David Hermann (2004)

Bulletin de la Société Mathématique de France

The aim of this paper is to compare two symplectic capacities in ℂ n related with periodic orbits of Hamiltonian systems: the Floer-Hofer capacity arising from symplectic homology, and the Viterbo capacity based on generating functions. It is shown here that the inner Floer-Hofer capacity is not larger than the Viterbo capacity and that they are equal for open sets with restricted contact type boundary. As an application, we prove that the Viterbo capacity of any compact Lagrangian submanifold is...

Integrability of Jacobi and Poisson structures

Marius Crainic, Chenchang Zhu (2007)

Annales de l’institut Fourier

We discuss the integrability of Jacobi manifolds by contact groupoids, and then look at what the Jacobi point of view brings new into Poisson geometry. In particular, using contact groupoids, we prove a Kostant-type theorem on the prequantization of symplectic groupoids, which answers a question posed by Weinstein and Xu. The methods used are those of Crainic-Fernandes on A -paths and monodromy group(oid)s of algebroids. In particular, most of the results we obtain are valid also in the non-integrable...

Integrable hierarchies and the modular class

Pantelis A. Damianou, Rui Loja Fernandes (2008)

Annales de l’institut Fourier

It is well-known that the Poisson-Nijenhuis manifolds, introduced by Kosmann-Schwarzbach and Magri form the appropriate setting for studying many classical integrable hierarchies. In order to define the hierarchy, one usually specifies in addition to the Poisson-Nijenhuis manifold a bi-hamiltonian vector field. In this paper we show that to every Poisson-Nijenhuis manifold one can associate a canonical vector field (no extra choices are involved!) which under an appropriate assumption defines an...

Integrable systems and group actions

Eva Miranda (2014)

Open Mathematics

The main purpose of this paper is to present in a unified approach to different results concerning group actions and integrable systems in symplectic, Poisson and contact manifolds. Rigidity problems for integrable systems in these manifolds will be explored from this perspective.

Integral formulas related to wave fronts

Sergeĭ Anisov (1999)

Banach Center Publications

In the first section of the paper we study some properties of oriented volumes of wave fronts propagating in spaces of constant curvature. In the second section, we generalize to an arbitrary isometric action of a Lie group on a Riemannian manifold the following principle: an extra pression inside of a ball does not move it.

Invariance of global solutions of the Hamilton-Jacobi equation

Ezequiel Maderna (2002)

Bulletin de la Société Mathématique de France

We show that every global viscosity solution of the Hamilton-Jacobi equation associated with a convex and superlinear Hamiltonian on the cotangent bundle of a closed manifold is necessarily invariant under the identity component of the group of symmetries of the Hamiltonian (we prove that this group is a compact Lie group). In particular, every Lagrangian section invariant under the Hamiltonian flow is also invariant under this group.

Invariants of real symplectic four-manifolds out of reducible and cuspidal curves

Jean-Yves Welschinger (2006)

Bulletin de la Société Mathématique de France

We construct invariants under deformation of real symplectic four-manifolds. These invariants are obtained by counting three different kinds of real rational J -holomorphic curves which realize a given homology class and pass through a given real configuration of (the appropriate number of) points. These curves are cuspidal curves, reducible curves and curves with a prescribed tangent line at some real point of the configuration. They are counted with respect to some sign defined by the parity of...

Invariants symplectiques et semi-classiques des systèmes intégrables avec singularités

San Vũ Ngọc (2000/2001)

Séminaire Équations aux dérivées partielles

On définit les notions de feuilletages classiques et semi-classiques pour les systèmes complètement intégrables avec singularités. Les résultats de classification standard (telles les coordonnées actions-angles semi-classiques) sont rappelés. Le cas du feuilletage classique de type foyer-foyer est examiné en détail, où des nouveaux invariants semi-globaux apparaissent. Ces invariants sont identifiés dans les conditions de Bohr-Sommerfeld singulières qui donnent le spectre conjoint au voisinage d’une...

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