scholarly journals Rigid subsets of symplectic manifolds

2009 ◽  
Vol 145 (03) ◽  
pp. 773-826 ◽  
Author(s):  
Michael Entov ◽  
Leonid Polterovich
Keyword(s):  
Symplectic Manifold ◽  
Lagrangian Submanifolds ◽  
Symplectic Manifolds ◽  
Smooth Functions ◽  
Torus Actions ◽  
Moment Maps ◽  
Quantum Homology ◽  
Singular Sets ◽  
Hamiltonian Torus Actions ◽  
Commutative Subalgebras

AbstractWe show that there is an hierarchy of intersection rigidity properties of sets in a closed symplectic manifold: some sets cannot be displaced by symplectomorphisms from more sets than the others. We also find new examples of rigidity of intersections involving, in particular, specific fibers of moment maps of Hamiltonian torus actions, monotone Lagrangian submanifolds (following the works of P. Albers and P. Biran-O. Cornea) as well as certain, possibly singular, sets defined in terms of Poisson-commutative subalgebras of smooth functions. In addition, we get some geometric obstructions to semi-simplicity of the quantum homology of symplectic manifolds. The proofs are based on the Floer-theoretical machinery of partial symplectic quasi-states.

scholarly journals Convexity of the moment map image for torus actions on b m -symplectic manifolds

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170420 ◽  
Author(s):  
Victor W. Guillemin ◽  
Eva Miranda ◽  
Jonathan Weitsman
Keyword(s):  
Symplectic Manifold ◽  
Integrable Systems ◽  
Symplectic Manifolds ◽  
Moment Map ◽  
Torus Action ◽  
Convexity Theorem ◽  
Theme Issue ◽  
Torus Actions ◽  
Finite Dimensional ◽  
The Moment

We prove a convexity theorem for the image of the moment map of a Hamiltonian torus action on a b m -symplectic manifold. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

scholarly journals Convexity for Hamiltonian torus actions on $b$-symplectic manifolds

2017 ◽  
Vol 24 (2) ◽  
pp. 363-377 ◽  
Author(s):  
Victor Guillemin ◽  
Eva Miranda ◽  
Ana Rita Pires ◽  
Geoffrey Scott
Keyword(s):  
Symplectic Manifolds ◽  
Torus Actions ◽  
Hamiltonian Torus Actions

scholarly journals Existence of pseudoheavy fibers of moment maps

2020 ◽  
pp. 2050047
Author(s):  
Morimichi Kawasaki ◽  
Ryuma Orita
Keyword(s):  
Lagrangian Submanifolds ◽  
Symplectic Manifolds ◽  
Partial Answer ◽  
Moment Maps

In this paper, we introduce the notion of pseudoheaviness of closed subsets of closed symplectic manifolds and prove the existence of pseudoheavy fibers of moment maps. In particular, we generalize Entov and Polterovich’s theorem, which ensures the existence of non-displaceable fibers. As its application, we provide a partial answer to a problem posed by them, which asks the existence of heavy fibers. Moreover, we obtain a family of singular Lagrangian submanifolds in [Formula: see text] with various rigidities.

Some Properties of Symplectic Manifolds S on the Projective Tangent Bundle PTM

2011 ◽  
Vol 187 ◽  
pp. 483-486
Author(s):  
Yong He ◽  
Xiao Ying Lu ◽  
Wei Na Lu
Keyword(s):  
Vector Field ◽  
Symplectic Manifold ◽  
Tangent Bundle ◽  
Fiber Bundle ◽  
Symplectic Manifolds ◽  
Lie Derivative ◽  
The Relationship

In this paper, we show the relationship between 2-form of the two projective tangent bundle and the relationship between 2-form on projective tangent bundle and 1-form on by using the theory of fiber bundle and the properties of symplectic manifold of the projective tangent bundle . Moreover, we derived a simpler formula of Lie derivative of a special vector field, which is on the projective tangent bundle.

scholarly journals A COMPARISON OF HOFER'S METRICS ON HAMILTONIAN DIFFEOMORPHISMS AND LAGRANGIAN SUBMANIFOLDS

2003 ◽  
Vol 05 (05) ◽  
pp. 803-811 ◽  
Author(s):  
YARON OSTROVER
Keyword(s):  
Symplectic Manifold ◽  
Isometric Embedding ◽  
Lagrangian Submanifolds ◽  
Canonical Embedding ◽  
Hamiltonian Diffeomorphisms

We compare Hofer's geometries on two spaces associated with a closed symplectic manifold (M,ω). The first space is the group of Hamiltonian diffeomorphisms. The second space ℒ consists of all Lagrangian submanifolds of M × M which are exact Lagrangian isotopic to the diagonal. We show that in the case of a closed symplectic manifold with π2(M) = 0, the canonical embedding of Ham (M) into ℒ, f ↦ graph (f) is not an isometric embedding, although it preserves Hofer's length of smooth paths.

scholarly journals Minimal Maslov number of R-spaces canonically embedded in Einstein-Kähler C-spaces

2019 ◽  
Vol 6 (1) ◽  
pp. 303-319
Author(s):  
Yoshihiro Ohnita
Keyword(s):  
Symmetric Space ◽  
Floer Homology ◽  
Lagrangian Submanifolds ◽  
Symplectic Manifolds ◽  
Lagrangian Submanifold ◽  
Riemannian Symmetric Space ◽  
Canonical Embedding ◽  
Real Form ◽  
Isotropy Representation ◽  
Totally Geodesic

AbstractAn R-space is a compact homogeneous space obtained as an orbit of the isotropy representation of a Riemannian symmetric space. It is known that each R-space has the canonical embedding into a Kähler C-space as a real form, and thus a compact embedded totally geodesic Lagrangian submanifold. The minimal Maslov number of Lagrangian submanifolds in symplectic manifolds is one of invariants under Hamiltonian isotopies and very fundamental to study the Floer homology for intersections of Lagrangian submanifolds. In this paper we show a Lie theoretic formula for the minimal Maslov number of R-spaces canonically embedded in Einstein-Kähler C-spaces, and provide some examples of the calculation by the formula.

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