Hamiltonian actions and Lagrangian homogeneous submanifolds (original) (raw)
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Hamiltonian actions and homogeneous Lagrangian submanifolds
Tohoku Mathematical Journal, 2007
We consider a connected symplectic manifold M acted on properly and in a Hamiltonian fashion by a connected Lie group G. Inspired to the recent paper [3], see also [12] and [24], we study Lagrangian orbits of Hamiltonian actions. The dimension of the moduli space of the Lagrangian orbits is given and also we describe under which condition a Lagrangian orbit is isolated. If M is a compact Kähler manifold we give a necessary and sufficient condition to an isometric action admits a Lagrangian orbit. Then we investigate Lagrangian homogeneous submanifolds on the symplectic cut and on the symplectic reduction. As an application of our results, we give new examples of Lagrangian homogeneous submanifolds on the blow-up at one point of the projective space and on the weighted projective spaces. Finally, applying Proposition 3.7 that we may call Lagrangian slice theorem for group acting with a fixed point, we give new examples of Lagrangian homogeneous submanifolds on irreducible Hermitian symmetric spaces of compact and noncompact type.
On geometric properties of Lagrangian submanifolds in product symplectic spaces
Hokkaido Mathematical Journal, 2006
We study the generic properties of symplectic relations. Local models of symplectic relations are described and the corresponding local symplectic invariants are derived. A stratification of the Lagrangian Grassmannian in the product symplectic space (N × M, π * M ω M − π * N ω N) is constructed and global homological properties of the strata are investigated.
Lagrangian submanifolds in product symplectic spaces
Journal of Mathematical Physics, 2000
We analyze the global structure of Lagrangian Grassmannian in the product symplectic space and investigate the local properties of generic symplectic relations. The cohomological symplectic invariant of discrete dynamical systems is generalized to the class of generalized canonical mappings. Lower bounds for the number of two-point and three-point symplectic invariants for billiard-type dynamical systems are found and several examples of symplectic correspondences encountered from physics are presented.
The reduced spaces of a symplectic Lie group action
Annals of Global Analysis and Geometry, 2006
There exist three main approaches to reduction associated to canonical Lie group actions on a symplectic manifold, namely, foliation reduction, introduced by Cartan, Marsden-Weinstein reduction, and optimal reduction, introduced by the authors. When the action is free, proper, and admits a momentum map these three approaches coincide. The goal of this paper is to study the general case of a symplectic action that does not admit a momentum map and one needs to use its natural generalization, a cylinder valued momentum map introduced by Condevaux, Dazord, and Molino . In this case it will be shown that the three reduced spaces mentioned above do not coincide, in general. More specifically, the Marsden-Weinstein reduced spaces are not symplectic but Poisson and their symplectic leaves are given by the optimal reduced spaces. Foliation reduction produces a symplectic reduced space whose Poisson quotient by a certain Lie group associated to the group of symmetries of the problem equals the Marsden-Weinstein reduced space. We illustrate these constructions with concrete examples, special emphasis being given to the reduction of a magnetic cotangent bundle of a Lie group in the situation when the magnetic term ensures the non-existence of the momentum map for the lifted action. The precise relation of the cylinder valued momentum map with group valued momentum maps for Abelian Lie groups is also given.
Lagrangian submanifolds in k-symplectic settings
Monatshefte für Mathematik, 2013
In this paper we extend the well-know normal form theorem for Lagrangian submanifolds proved by A. Weinstein in symplectic geometry to the setting of k-symplectic manifolds.
The stratified spaces of a symplectic lie group action
Reports on Mathematical Physics, 2006
There exist three main approaches to reduction associated to canonical Lie group actions on a symplectic manifold, namely, foliation reduction, introduced by Cartan, Marsden-Weinstein reduction, and optimal reduction, introduced by the authors. When the action is free, proper, and admits a momentum map these three approaches coincide. The goal of this paper is to study the general case of a symplectic action that does not admit a momentum map and one needs to use its natural generalization, a cylinder valued momentum map introduced by Condevaux, Dazord, and Molino . In this case it will be shown that the three reduced spaces mentioned above do not coincide, in general. More specifically, the Marsden-Weinstein reduced spaces are not symplectic but Poisson and their symplectic leaves are given by the optimal reduced spaces. Foliation reduction produces a symplectic reduced space whose Poisson quotient by a certain Lie group associated to the group of symmetries of the problem equals the Marsden-Weinstein reduced space. We illustrate these constructions with concrete examples, special emphasis being given to the reduction of a magnetic cotangent bundle of a Lie group in the situation when the magnetic term ensures the non-existence of the momentum map for the lifted action. The precise relation of the cylinder valued momentum map with group valued momentum maps for Abelian Lie groups is also given.
Balkan Journal of Geometry and Its Applications
The aim of this paper is to give a positive answer to the following question concerning the Hamiltonian submanifolds: for a given Hamiltonian, does it exist a section of the projection of cotangent bundles, which depends only on the Hamiltonian? is to show that a natural distinguished sectioñ i exists and it depends only on the Hamiltonian H. It implies that, under natural conditions, the Hamiltonian H on the manifold M induces the Hamiltonian H = H' on the submanifold M' . This construction is performed in a different way in [8], using the Lagrangian and the Hamiltonian formalisms. In the first section we recall briefly some classical results used in the paper, con-cerning Legendre transformations. In the second section we construct explicitly the sectioñ i, using the Legendre transformation as an essential ingredient.
European Journal of Mathematics
Let M be a holomorphic symplectic Kähler manifold equipped with a Lagrangian fibration π with compact fibers. The base of this manifold is equipped with a special Kähler structure, that is, a Kähler structure (I, g, ω) and a symplectic flat connection ∇ such that the metric g is locally the Hessian of a function. We prove that any Lagrangian subvariety Z ⊂ M which intersects smooth fibers of π and smoothly projects to π(Z) is a toric fibration over its image π(Z) in B, and this image is also special Kähler. This answers a question of N. Hitchin related to Kapustin-Witten BBB/BAA duality.
SYMPLECTIC QUOTIENTS AND COMPLEX GEOMETRY
In this memoire we present the theory of symplectic quotients and the interactions with the theory of kahler quotients. The starting point is the following natural question: in the presence of a symplectic action of a Lie group G on a symplectic manifold pM, ωq can one define in a coherent way the quotient of M by G (so that the result will have a natural symplectic structure)? Even if we assume that G is compact and the action is free, we see that in general M {G cannot be symplectic (for dimension reasons). In order to obtain a symplectic quotient, one needs a new ingredient: a moment map. The fundamental theorem of the theory states that, assuming that G acts freely and properly around the zero locus of the moment map, then the G-quotient of this zero locus has a natural structure of a symplectic manifold. We will solve in detail the existence and unicity problems for moment maps. The main tool needed here is Lie algebra cohomology. We will continue with explicit computations of moment maps and explicit descriptions of symplectic quotients. We will prove that many interesting manifolds (projective spaces, Grassman manifolds, flag manifolds) can be obtained as symplectic quotients. We will conclude with a general principle which emphasizes an interesting relation between symplectic geometry and complex geometry: Let X be a Kähler manifold, K be a compact Lie group and G " K CˆX Ñ X be a holomorphic action on X which restricts to a Hamiltonian symplectic action of K which is free around µ´1p0q. Then the corresponding symplectic quotient has a natural structure of complex manifold. We will illustrate this principle in the examples we study.
Remarks on Symplectic Geometry
arXiv: Symplectic Geometry, 2019
We survey the progresses on the study of symplectic geometry past four decades. We briefly deal with the convexity properties of a moment map, the classification of symplectic actions, the symplectic embedding problems, and the theory of Gromov-Witten invariants.