Characterization of diffeomorphisms that are symplectomorphisms

Stanisław Janeczko; Zbigniew Jelonek

Displaying similar documents to “Characterization of diffeomorphisms that are symplectomorphisms”

Special Lagrangian linear subspaces in product symplectic space

Małgorzata Mikosz (2004)

Banach Center Publications

Similarity:

The notes consist of a study of special Lagrangian linear subspaces. We will give a condition for the graph of a linear symplectomorphism $f : (ℝ^{2 n}, σ = \sum_{i = 1}^{n} d x_{i} \land d y_{i}) \to (ℝ^{2 n}, σ)$ to be a special Lagrangian linear subspace in $(ℝ^{2 n} \times ℝ^{2 n}, ω = π * ₂ σ - π * ₁ σ)$ . This way a special symplectic subset in the symplectic group is introduced. A stratification of special Lagrangian Grassmannian $S Λ_{2 n} ≃ S U (2 n) / S O (2 n)$ is defined.

Symplectic periods of the continuous spectrum of $GL (2 n)$

Shunsuke Yamana (2014)

Annales de l’institut Fourier

Similarity:

We provide a formula for the symplectic period of an Eisenstein series on $GL (2 n)$ and determine when it is not identically zero.

$𝒞^{0}$ -rigidity of characteristics in symplectic geometry

Emmanuel Opshtein (2009)

Annales scientifiques de l'École Normale Supérieure

Similarity:

The paper concerns a $𝒞^{0}$ -rigidity result for the characteristic foliations in symplectic geometry. A symplectic homeomorphism (in the sense of Eliashberg-Gromov) which preserves a smooth hypersurface also preserves its characteristic foliation.

Generalized Conley-Zehnder index

Jean Gutt (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

Similarity:

The Conley-Zehnder index associates an integer to any continuous path of symplectic matrices starting from the identity and ending at a matrix which does not admit $1$ as an eigenvalue. Robbin and Salamon define a generalization of the Conley-Zehnder index for any continuous path of symplectic matrices; this generalization is half integer valued. It is based on a Maslov-type index that they define for a continuous path of Lagrangians in a symplectic vector space $(W, \bar{Ω})$ , having chosen a given...

A characterization of symplectic groups related to Fermat primes

Behnam Ebrahimzadeh, Alireza K. Asboei (2021)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We proved that the symplectic groups $PSp (4, 2^{n})$ , where $2^{2 n} + 1$ is a Fermat prime number is uniquely determined by its order, the first largest element orders and the second largest element orders.

Rabinowitz Floer homology and symplectic homology

Kai Cieliebak, Urs Frauenfelder, Alexandru Oancea (2010)

Annales scientifiques de l'École Normale Supérieure

Similarity:

The first two authors have recently defined Rabinowitz Floer homology groups $R F H_{*} (M, W)$ associated to a separating exact embedding of a contact manifold $(M, ξ)$ into a symplectic manifold $(W, ω)$ . These depend only on the bounded component $V$ of $W ∖ M$ . We construct a long exact sequence in which symplectic cohomology of $V$ maps to symplectic homology of $V$ , which in turn maps to Rabinowitz Floer homology $R F H_{*} (M, W)$ , which then maps to symplectic cohomology of $V$ . We compute $R F H_{*} (S T^{*} L, T^{*} L)$ , where $S T^{*} L$ is the unit cosphere bundle of a closed...

Moduli spaces of polarized irreducible symplectic manifolds are not necessarily connected

Apostol Apostolov (2014)

Annales de l’institut Fourier

Similarity:

We show that the moduli space of polarized irreducible symplectic manifolds of $K 3^{[n]}$ -type, of fixed polarization type, is not always connected. This can be derived as a consequence of Eyal Markman’s characterization of polarized parallel-transport operators of $K 3^{[n]}$ -type.

Hofer’s metrics and boundary depth

Michael Usher (2013)

Annales scientifiques de l'École Normale Supérieure

Similarity:

We show that if $(M, ω)$ is a closed symplectic manifold which admits a nontrivial Hamiltonian vector field all of whose contractible closed orbits are constant, then Hofer’s metric on the group of Hamiltonian diffeomorphisms of $(M, ω)$ has infinite diameter, and indeed admits infinite-dimensional quasi-isometrically embedded normed vector spaces. A similar conclusion applies to Hofer’s metric on various spaces of Lagrangian submanifolds, including those Hamiltonian-isotopic to the diagonal in $M \times M$ ...

Rational symplectic field theory over $ℤ^{2}$ for exact Lagrangian cobordisms

Tobias Ekholm (2008)

Journal of the European Mathematical Society

Similarity:

We construct a version of rational Symplectic Field Theory for pairs $(X, L)$ , where $X$ is an exact symplectic manifold, where $L \subset X$ is an exact Lagrangian submanifold with components subdivided into $k$ subsets, and where both $X$ and $L$ have cylindrical ends. The theory associates to $(X, L)$ a $ℤ$ -graded chain complex of vector spaces over $ℤ_{2}$ , filtered with $k$ filtration levels. The corresponding $k$ -level spectral sequence is invariant under deformations of $(X, L)$ and has the following property: if $(X, L)$ is obtained by...

Symplectic critical surfaces in Kähler surfaces

Xiaoli Han, Jiayu Li (2010)

Journal of the European Mathematical Society

Similarity:

Let $M$ be a Kähler surface and $Σ$ be a closed symplectic surface which is smoothly immersed in $M$ . Let $α$ be the Kähler angle of $Σ$ in $M$ . We first deduce the Euler-Lagrange equation of the functional $L = \int_{Σ} \frac{1}{cos α} d μ$ in the class of symplectic surfaces. It is ${cos}^{3} α H = {(J {(J \nabla cos α)}^{⊤})}^{⊥}$ , where $H$ is the mean curvature vector of $Σ$ in $M$ , $J$ is the complex structure compatible with the Kähler form $ω$ in $M$ , which is an elliptic equation. We call such a surface a symplectic critical surface. We show that, if $M$ is a Kähler-Einstein surface...

Some lagrangian invariants of symplectic manifolds

Michel Nguiffo Boyom (2007)

Banach Center Publications

Similarity:

The KV-homology theory is a new framework which yields interesting properties of lagrangian foliations. This short note is devoted to relationships between the KV-homology and the KV-cohomology of a lagrangian foliation. Let us denote by $_{F}$ (resp. $V^{F}$ ) the KV-algebra (resp. the space of basic functions) of a lagrangian foliation F. We show that there exists a pairing of cohomology and homology to $V^{F}$ . That is to say, there is a bilinear map $H^{q} (_{F}, V^{F}) \times H_{q} (_{F}, V^{F}) \to V^{F}$ , which is invariant under F-preserving symplectic...

A vanishing theorem for twisted Alexander polynomials with applications to symplectic 4-manifolds

Stefan Friedl, Stefano Vidussi (2013)

Journal of the European Mathematical Society

Similarity:

In this paper we show that given any 3-manifold $N$ and any non-fibered class in $H^{1} (N; Z)$ there exists a representation such that the corresponding twisted Alexander polynomial is zero. We obtain this result by extending earlier work of ours and by combining this with recent results of Agol and Wise on separability of 3-manifold groups. This result allows us to completely classify symplectic 4-manifolds with a free circle action, and to determine their symplectic cones.

Exotic Deformations of Calabi-Yau Manifolds

Paolo de Bartolomeis, Adriano Tomassini (2013)

Annales de l’institut Fourier

Similarity:

We introduce Quantum Inner State manifolds (QIS manifolds) as (compact) $2 n$ -dimensional symplectic manifolds $(M, κ)$ endowed with a $κ$ -tamed almost complex structure $J$ and with a nowhere vanishing and normalized section $ϵ$ of the bundle $Λ_{J}^{n, 0} (M)$ satisfying the condition ${\bar{\partial}}_{J} ϵ = 0$ . We study the moduli space $𝔐$ of QIS deformations of a given Calabi-Yau manifold, computing its tangent space...

The Lie groupoid analogue of a symplectic Lie group

David N. Pham (2021)

Archivum Mathematicum

Similarity:

A symplectic Lie group is a Lie group with a left-invariant symplectic form. Its Lie algebra structure is that of a quasi-Frobenius Lie algebra. In this note, we identify the groupoid analogue of a symplectic Lie group. We call the aforementioned structure a $t$ -symplectic Lie groupoid; the “ $t$ " is motivated by the fact that each target fiber of a $t$ -symplectic Lie groupoid is a symplectic manifold. For a Lie groupoid $𝒢 ⇉ M$ , we show that there is a one-to-one correspondence between quasi-Frobenius...

Projective structure, $\tilde{SL} (3, ℝ)$ and the symplectic Dirac operator

Marie Holíková, Libor Křižka, Petr Somberg (2016)

Archivum Mathematicum

Similarity:

Inspired by the results on symmetries of the symplectic Dirac operator, we realize symplectic spinor fields and the symplectic Dirac operator in the framework of (the double cover of) homogeneous projective structure in two real dimensions. The symmetry group of the homogeneous model of the double cover of projective geometry in two real dimensions is $\tilde{} (3,)$ .

Robust transitivity in hamiltonian dynamics

Meysam Nassiri, Enrique R. Pujals (2012)

Annales scientifiques de l'École Normale Supérieure

Similarity:

A goal of this work is to study the dynamics in the complement of KAM tori with focus on non-local robust transitivity. We introduce $C^{r}$ open sets ( $r = 1, 2, \dots, \infty$ ) of symplectic diffeomorphisms and Hamiltonian systems, exhibitingrobustly transitive sets. We show that the $C^{\infty}$ closure of such open sets contains a variety of systems, including so-called unstable integrable systems. In addition, the existence of ergodic measures with large support is obtained for all those systems. A main ingredient of the...

Lagrangian fibrations on generalized Kummer varieties

Martin G. Gulbrandsen (2007)

Bulletin de la Société Mathématique de France

Similarity:

We investigate the existence of Lagrangian fibrations on the generalized Kummer varieties of Beauville. For a principally polarized abelian surface $A$ of Picard number one we find the following: The Kummer variety $K^{n} A$ is birationally equivalent to another irreducible symplectic variety admitting a Lagrangian fibration, if and only if $n$ is a perfect square. And this is the case if and only if $K^{n} A$ carries a divisor with vanishing Beauville-Bogomolov square.