scholarly journals Contact Dynamics: Legendrian and Lagrangian Submanifolds

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2704
Author(s):  
Oğul Esen ◽  
Manuel Lainz Valcázar ◽  
Manuel de León ◽  
Juan Carlos Marrero
Keyword(s):  
Lagrangian Submanifolds ◽  
Contact Manifold ◽  
Symplectic Manifolds ◽  
Contact Dynamics ◽  
Legendre Transformation ◽  
Lagrangian Dynamics ◽  
Symplectic Diffeomorphisms ◽  
Legendrian Submanifold ◽  
Legendrian Submanifolds

We are proposing Tulczyjew’s triple for contact dynamics. The most important ingredients of the triple, namely symplectic diffeomorphisms, special symplectic manifolds, and Morse families, are generalized to the contact framework. These geometries permit us to determine so-called generating family (obtained by merging a special contact manifold and a Morse family) for a Legendrian submanifold. Contact Hamiltonian and Lagrangian Dynamics are recast as Legendrian submanifolds of the tangent contact manifold. In this picture, the Legendre transformation is determined to be a passage between two different generators of the same Legendrian submanifold. A variant of contact Tulczyjew’s triple is constructed for evolution contact dynamics.

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.

scholarly journals TRANSVERSAL HARMONIC THEORY FOR TRANSVERSALLY SYMPLECTIC FLOWS

2008 ◽  
Vol 84 (2) ◽  
pp. 233-245 ◽  
Author(s):  
HONG KYUNG PAK
Keyword(s):  
Compact Manifold ◽  
Hodge Structure ◽  
Contact Manifold ◽  
Symplectic Manifolds ◽  
Poisson Manifold ◽  
Harmonic Forms ◽  
Cosymplectic Manifolds ◽  
Almost Cosymplectic Manifold ◽  
Differential Complex ◽  
Hard Lefschetz Theorem

AbstractWe develop the transversal harmonic theory for a transversally symplectic flow on a manifold and establish the transversal hard Lefschetz theorem. Our main results extend the cases for a contact manifold (H. Kitahara and H. K. Pak, ‘A note on harmonic forms on a compact manifold’, Kyungpook Math. J.43 (2003), 1–10) and for an almost cosymplectic manifold (R. Ibanez, ‘Harmonic cohomology classes of almost cosymplectic manifolds’, Michigan Math. J.44 (1997), 183–199). For the point foliation these are the results obtained by Brylinski (‘A differential complex for Poisson manifold’, J. Differential Geom.28 (1988), 93–114), Haller (‘Harmonic cohomology of symplectic manifolds’, Adv. Math.180 (2003), 87–103), Mathieu (‘Harmonic cohomology classes of symplectic manifolds’, Comment. Math. Helv.70 (1995), 1–9) and Yan (‘Hodge structure on symplectic manifolds’, Adv. Math.120 (1996), 143–154).

scholarly journals Reduced dynamics and Lagrangian submanifolds of symplectic manifolds

2014 ◽  
Vol 47 (22) ◽  
pp. 225203 ◽  
Author(s):  
E García-Toraño Andrés ◽  
E Guzmán ◽  
J C Marrero ◽  
T Mestdag
Keyword(s):  
Lagrangian Submanifolds ◽  
Symplectic Manifolds

scholarly journals Differential Invariants of Measurements, and Their Relation to Central Moments

Entropy ◽  
2020 ◽  
Vol 22 (10) ◽  
pp. 1118
Author(s):  
Eivind Schneider
Keyword(s):  
Affine Space ◽  
Information Gain ◽  
Probability Distributions ◽  
Differential Invariants ◽  
Affine Transformations ◽  
Equilibrium Thermodynamics ◽  
Central Moments ◽  
Legendrian Submanifold ◽  
Legendrian Submanifolds ◽  
Minimal Information

Due to the principle of minimal information gain, the measurement of points in an affine space V determines a Legendrian submanifold of V×V*×R. Such Legendrian submanifolds are equipped with additional geometric structures that come from the central moments of the underlying probability distributions and are invariant under the action of the group of affine transformations on V. We investigate the action of this group of affine transformations on Legendrian submanifolds of V×V*×R by giving a detailed overview of the structure of the algebra of scalar differential invariants, and we show how the scalar differential invariants can be constructed from the central moments. In the end, we view the results in the context of equilibrium thermodynamics of gases, and notice that the heat capacity is one of the differential invariants.

scholarly journals ON ITERATED TRANSLATED POINTS FOR CONTACTOMORPHISMS OF ℝ2n+1 AND ℝ2n × S1

2012 ◽  
Vol 23 (02) ◽  
pp. 1250042 ◽  
Author(s):  
SHEILA SANDON
Keyword(s):  
Fixed Points ◽  
Critical Points ◽  
Generating Functions ◽  
Contact Manifold ◽  
Contact Form ◽  
Compactly Supported ◽  
Partial Contact ◽  
Legendrian Submanifolds ◽  
Special Case

A point q in a contact manifold is called a translated point for a contactomorphism ϕ with respect to some fixed contact form if ϕ(q) and q belong to the same Reeb orbit and the contact form is preserved at q. The problem of existence of translated points has an interpretation in terms of Reeb chords between Legendrian submanifolds, and can be seen as a special case of the problem of leafwise coisotropic intersections. For a compactly supported contactomorphism ϕ of ℝ2n+1 or ℝ2n × S1 contact isotopic to the identity, existence of translated points follows immediately from Chekanov's theorem on critical points of quasi-functions and Bhupal's graph construction. In this article we prove that if ϕ is positive then there are infinitely many nontrivial geometrically distinct iterated translated points, i.e. translated points of some iteration ϕk. This result can be seen as a (partial) contact analog of the result of Viterbo on existence of infinitely many iterated fixed points for compactly supported Hamiltonian symplectomorphisms of ℝ2n, and is obtained with generating functions techniques.

scholarly journals Lefschetz–Bott Fibrations on Line Bundles Over Symplectic Manifolds

2020 ◽  
Author(s):  
Takahiro Oba
Keyword(s):  
Unit Disk ◽  
Blow Up ◽  
Contact Manifold ◽  
Symplectic Manifolds ◽  
Line Bundles ◽  
Complex Line ◽  
Open Book ◽  
Type Singularity ◽  
On Line ◽  
Symplectic Filling

Abstract We describe Lefschetz–Bott fibrations on complex line bundles over symplectic manifolds explicitly. As an application, we show that the link of the $A_{k}$-type singularity has more than one strong symplectic filling up to homotopy and blow-up at points when the dimension of the link is greater than or equal to $5$. In the appendix, we show that the total space of a Lefschetz–Bott fibration over the unit disk serves as a strong symplectic filling of a contact manifold compatible with an open book induced by the fibration.

scholarly journals Pseudosymmetry properties of generalised Wintgen ideal Legendrian submanifolds

Filomat ◽  
2019 ◽  
Vol 33 (4) ◽  
pp. 1209-1215
Author(s):  
Aleksandar Sebekovic ◽  
Miroslava Petrovic-Torgasev ◽  
Anica Pantic
Keyword(s):  
Curvature Tensor ◽  
Ricci Tensor ◽  
Space Forms ◽  
Equality Case ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Sasakian Space Forms ◽  
Legendrian Submanifold ◽  
Legendrian Submanifolds ◽  
Weyl Conformal Curvature Tensor

For Legendrian submanifolds Mn in Sasakian space forms ?M2n+1(c), I. Mihai obtained an inequality relating the normalised scalar curvature (intrinsic invariant) and the squared mean curvature and the normalised scalar normal curvature of M in the ambient space ?M (extrinsic invariants) which is called the generalised Wintgen inequality, characterising also the corresponding equality case. And a Legendrian submanifold Mn in Sasakian space forms ?M2n+1(c) is said to be generalised Wintgen ideal Legendrian submanifold of ?M2n+1(c) when it realises at everyone of its points the equality in such inequality. Characterisations based on some basic intrinsic symmetries involving the Riemann-Cristoffel curvature tensor, the Ricci tensor and the Weyl conformal curvature tensor belonging to the class of pseudosymmetries in the sense of Deszcz of such generalised Wintgen ideal Legendrian submanifolds are given.

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 Topological contact dynamics I: symplectization and applications of the energy-capacity inequality

2015 ◽  
Vol 15 (3) ◽  
Author(s):  
Stefan Müller ◽  
Peter Spaeth
Keyword(s):  
Smooth Manifold ◽  
Contact Structure ◽  
Measure Theory ◽  
Contact Manifold ◽  
Contact Dynamics ◽  
Energy Capacity ◽  
Uniqueness Results ◽  
Group Identities ◽  
Smooth Contact ◽  
Topological Automorphism

AbstractWe introduce topological contact dynamics of a smooth manifold carrying a cooriented contact structure, generalizing previous work in the case of a symplectic structure [27] or a contact form [5]. A topological contact isotopy is not generated by a vector field; nevertheless, the group identities, the transformation law, and classical uniqueness results in the smooth case extend to topological contact isotopies and homeomorphisms, giving rise to an extension of smooth contact dynamics to topological dynamics. Our approach is via symplectization of a contact manifold, and our main tools are an energy-capacity inequality we prove for contact diffeomorphisms, combined with techniques from measure theory on oriented manifolds. We establish non-degeneracy of a Hofer-like bi-invariant pseudo-metric on the group of strictly contact diffeomorphisms constructed in [4]. The topological automorphism group of the contact structure exhibits rigidity properties analogous to those of symplectic diffeomorphisms, including C

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