Displaying similar documents to “ 𝒞 0 -rigidity of characteristics in symplectic geometry”

Characterization of diffeomorphisms that are symplectomorphisms

Stanisław Janeczko, Zbigniew Jelonek (2009)

Fundamenta Mathematicae

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Let ( X , ω X ) and ( Y , ω Y ) be compact symplectic manifolds (resp. symplectic manifolds) of dimension 2n > 2. Fix 0 < s < n (resp. 0 < k ≤ n) and assume that a diffeomorphism Φ : X → Y maps all 2s-dimensional symplectic submanifolds of X to symplectic submanifolds of Y (resp. all isotropic k-dimensional tori of X to isotropic tori of Y). We prove that in both cases Φ is a conformal symplectomorphism, i.e., there is a constant c ≠0 such that Φ * ω Y = c ω X .

Special Lagrangian linear subspaces in product symplectic space

Małgorzata Mikosz (2004)

Banach Center Publications

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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 = ∑ i = 1 n d x i ∧ d y i ) → ( ℝ 2 n , σ ) to be a special Lagrangian linear subspace in ( ℝ 2 n × 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.

Generalized Conley-Zehnder index

Jean Gutt (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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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 , Ω ¯ ) , having chosen a given...

Projective structure, SL ˜ ( 3 , ℝ ) and the symplectic Dirac operator

Marie Holíková, Libor Křižka, Petr Somberg (2016)

Archivum Mathematicum

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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 ˜ ( 3 , ) .

Some lagrangian invariants of symplectic manifolds

Michel Nguiffo Boyom (2007)

Banach Center Publications

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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 ) × H q ( F , V F ) → V F , which is invariant under F-preserving symplectic...

Rabinowitz Floer homology and symplectic homology

Kai Cieliebak, Urs Frauenfelder, Alexandru Oancea (2010)

Annales scientifiques de l'École Normale Supérieure

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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...

Symplectic critical surfaces in Kähler surfaces

Xiaoli Han, Jiayu Li (2010)

Journal of the European Mathematical Society

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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 = ∫ Σ 1 cos α d μ in the class of symplectic surfaces. It is cos 3 α H = ( J ( J ∇ 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...

Robust transitivity in hamiltonian dynamics

Meysam Nassiri, Enrique R. Pujals (2012)

Annales scientifiques de l'École Normale Supérieure

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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 , ⋯ , ∞ ) of symplectic diffeomorphisms and Hamiltonian systems, exhibitingrobustly transitive sets. We show that the C ∞ 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...

Moduli spaces of polarized irreducible symplectic manifolds are not necessarily connected

Apostol Apostolov (2014)

Annales de l’institut Fourier

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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.

On some completions of the space of hamiltonian maps

Vincent Humilière (2008)

Bulletin de la Société Mathématique de France

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In one of his papers, C. Viterbo defined a distance on the set of Hamiltonian diffeomorphisms of ℝ 2 n endowed with the standard symplectic form ω 0 = d p ∧ d q . We study the completions of this space for the topology induced by Viterbo’s distance and some others derived from it, we study their different inclusions and give some of their properties. In particular, we give a convergence criterion for these distances that allows us to prove that the completions contain non-ordinary elements, as for example,...

Hofer’s metrics and boundary depth

Michael Usher (2013)

Annales scientifiques de l'École Normale Supérieure

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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 × M ...

Rational symplectic field theory over ℤ 2 for exact Lagrangian cobordisms

Tobias Ekholm (2008)

Journal of the European Mathematical Society

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We construct a version of rational Symplectic Field Theory for pairs ( X , L ) , where X is an exact symplectic manifold, where L ⊂ 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...

Groups of C r , s -diffeomorphisms related to a foliation

Jacek Lech, Tomasz Rybicki (2007)

Banach Center Publications

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The notion of a C r , s -diffeomorphism related to a foliation is introduced. A perfectness theorem for the group of C r , s -diffeomorphisms is proved. A remark on C n + 1 -diffeomorphisms is given.

Lagrangian fibrations on generalized Kummer varieties

Martin G. Gulbrandsen (2007)

Bulletin de la Société Mathématique de France

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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.

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

Stefan Friedl, Stefano Vidussi (2013)

Journal of the European Mathematical Society

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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

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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 ∂ ¯ J ϵ = 0 . We study the moduli space 𝔐 of QIS deformations of a given Calabi-Yau manifold, computing its tangent space...

The Brauer category and invariant theory

Gustav I. Lehrer, R. B. Zhang (2015)

Journal of the European Mathematical Society

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A category of Brauer diagrams, analogous to Turaev’s tangle category, is introduced, a presentation of the category is given, and full tensor functors are constructed from this category to the category of tensor representations of the orthogonal group O ( V ) or the symplectic group Sp ( V ) over any field of characteristic zero. The first and second fundamental theorems of invariant theory for these classical groups are generalised to the category theoretic setting. The major outcome is that...

Minimal, rigid foliations by curves on ℂ ℙ n

Frank Loray, Julio C. Rebelo (2003)

Journal of the European Mathematical Society

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We prove the existence of minimal and rigid singular holomorphic foliations by curves on the projective space ℂ ℙ n for every dimension n ≥ 2 and every degree d ≥ 2 . Precisely, we construct a foliation ℱ which is induced by a homogeneous vector field of degree d , has a finite singular set and all the regular leaves are dense in the whole of ℂ ℙ n . Moreover, ℱ satisfies many additional properties expected from chaotic dynamics and is rigid in the following sense: if ℱ is conjugate to another holomorphic...

On the differential geometry of some classes of infinite dimensional manifolds

Maysam Maysami Sadr, Danial Bouzarjomehri Amnieh (2024)

Archivum Mathematicum

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Albeverio, Kondratiev, and Röckner have introduced a type of differential geometry, which we call lifted geometry, for the configuration space Γ X of any manifold X . The name comes from the fact that various elements of the geometry of Γ X are constructed via lifting of the corresponding elements of the geometry of X . In this note, we construct a general algebraic framework for lifted geometry which can be applied to various “infinite dimensional spaces” associated to X . In order to define...