Generating families for images of Lagrangian submanifolds and open swallowtails

1986 ◽  
Vol 100 (1) ◽  
pp. 91-107 ◽  
Author(s):  
Stanisław Janeczko
Keyword(s):  
Generating Functions ◽  
Cotangent Bundle ◽  
Symplectic Structure ◽  
Normal Forms ◽  
Lagrangian Submanifolds ◽  
Lagrangian Submanifold ◽  
Classification Theorem ◽  
Local Models ◽  
Binary Forms

SummaryIn this paper we study the symplectic relations appearing as the generalized cotangent bundle liftings of smooth stable mappings. Using this class of symplectic relations the classification theorem for generic (pre) images of lagrangian submanifolds is proved. The normal forms for the respective classified puilbacks and pushforwards are provided and the connections between the singularity types of symplectic relation, mapped lagrangian submanifold and singular image, are established. The notion of special symplectic triplet is introduced and the generic local models of such triplets are studied. We show that the open swallowtails are canonically represented as pushforwards of the appropriate regular lagrangian submanifolds. Using the SL2(ℝ) invariant symplectic structure of the space of binary forms of n appropriate dimension we derive the generating families for the open swallowtails and the respective generating functions for its regular resolutions.

scholarly journals Hyperkähler metrics near Lagrangian submanifolds and symplectic groupoids

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Maxence Mayrand
Keyword(s):  
Symplectic Manifold ◽  
Cotangent Bundle ◽  
Lagrangian Submanifolds ◽  
Poisson Manifold ◽  
Lagrangian Submanifold ◽  
Zero Section ◽  
Poisson Structures ◽  
Symplectic Structures ◽  
Symplectic Groupoids ◽  
Hyperkähler Metrics

Abstract The first part of this paper is a generalization of the Feix–Kaledin theorem on the existence of a hyperkähler metric on a neighborhood of the zero section of the cotangent bundle of a Kähler manifold. We show that the problem of constructing a hyperkähler structure on a neighborhood of a complex Lagrangian submanifold in a holomorphic symplectic manifold reduces to the existence of certain deformations of holomorphic symplectic structures. The Feix–Kaledin structure is recovered from the twisted cotangent bundle. We then show that every holomorphic symplectic groupoid over a compact holomorphic Poisson surface of Kähler type has a hyperkähler structure on a neighborhood of its identity section. More generally, we reduce the existence of a hyperkähler structure on a symplectic realization of a holomorphic Poisson manifold of any dimension to the existence of certain deformations of holomorphic Poisson structures adapted from Hitchin’s unobstructedness theorem.

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.

SOME RESULTS ON GENERALIZED CALABI–YAU MANIFOLDS

2006 ◽  
Vol 03 (05n06) ◽  
pp. 1273-1292
Author(s):  
PAOLO DE BARTOLOMEIS ◽  
ADRIANO TOMASSINI
Keyword(s):  
Lagrangian Submanifolds ◽  
Lagrangian Submanifold ◽  
Dimensional Case ◽  
Maslov Class ◽  
Calabi Yau Manifolds

We consider generalized Calabi–Yau manifolds and we give a formula for the Maslov class of a Lagrangian submanifold of a generalized Calabi–Yau manifold. In particular, we characterize the Lagrangian submanifolds with vanishing Maslov class. In the 6-dimensional case, we refine our definition. Finally, we construct some examples.

scholarly journals Hyperkähler metrics associated to compact Lie groups

1996 ◽  
Vol 120 (1) ◽  
pp. 61-69 ◽  
Author(s):  
Andrew Dancer ◽  
Andrew Swann
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Cotangent Bundle ◽  
Symplectic Structure ◽  
Compact Lie Group ◽  
Compact Lie Groups ◽  
Left And Right ◽  
Hyperkähler Metrics

It is well known that the cotangent bundle of any manifold has a canonical symplectic structure. If we specialize to the case when the manifold is a compact Lie group G, then this structure is preserved by the actions of G on T*G induced by left and right translation on G. We refer to these as the left and right actions of G on T*G.

scholarly journals Lagrangian Submanifolds of Symplectic Structures Induced by Divergence Functions

Entropy ◽  
2020 ◽  
Vol 22 (9) ◽  
pp. 983
Author(s):  
Marco Favretti
Keyword(s):  
Symplectic Structure ◽  
Probability Distributions ◽  
Lagrangian Submanifolds ◽  
Maximum Entropy Principle ◽  
Information Geometry ◽  
Dimensional Manifold ◽  
Finite Dimensional ◽  
Relevant Role ◽  
Entropy Principle ◽  
Dual Connection

Divergence functions play a relevant role in Information Geometry as they allow for the introduction of a Riemannian metric and a dual connection structure on a finite dimensional manifold of probability distributions. They also allow to define, in a canonical way, a symplectic structure on the square of the above manifold of probability distributions, a property that has received less attention in the literature until recent contributions. In this paper, we hint at a possible application: we study Lagrangian submanifolds of this symplectic structure and show that they are useful for describing the manifold of solutions of the Maximum Entropy principle.

ON LAGRANGIAN EMBEDDINGS OF PARALLELIZABLE MANIFOLDS

2013 ◽  
Vol 24 (09) ◽  
pp. 1350073
Author(s):  
NAOHIKO KASUYA ◽  
TORU YOSHIYASU
Keyword(s):  
Euclidean Space ◽  
Symplectic Structure ◽  
Lagrangian Submanifold ◽  
Symplectic Structures

We prove that for any closed parallelizable n-manifold Mn, if the dimension n ≠ 7, or if n = 7 and the Kervaire semi-characteristic χ½(M7) is zero, then Mn can be embedded in the Euclidean space ℝ2n with a certain symplectic structure as a Lagrangian submanifold. By the results of Gromov and Fukaya, our result gives rise to symplectic structures of ℝ2n(n ≥ 3) which are not conformally equivalent to open domains in standard ones.

SYMPLECTIC REDUCTION FOR YANG–MILLS ON A CYLINDER

2004 ◽  
Vol 01 (04) ◽  
pp. 289-298 ◽  
Author(s):  
AMBAR N. SENGUPTA
Keyword(s):  
Gauge Group ◽  
Cotangent Bundle ◽  
Symplectic Structure ◽  
Hamiltonian Dynamics ◽  
Symplectic Reduction ◽  
Yang Mills

An account of the Lagrangian and Hamiltonian dynamics of the pure Yang–Mills system is presented. This framework is applied to the case of (1+1)-dimensional cylindrical spacetime. Hamiltonian dynamics on the space of connections over a circle is often identified with dynamics on the cotangent bundle of the gauge group by means of the holonomy. In support of this procedure we show that the symplectic structure for Hamiltonian dynamics for connections on a circle is identifiable with the natural symplectic structure on the cotangent bundle of the gauge group.

Symplectic structure and gauge invariance on the cotangent bundle

1994 ◽  
Vol 35 (1) ◽  
pp. 426-434 ◽  
Author(s):  
L. Hernández Encinas ◽  
J. Muñoz Masqué
Keyword(s):  
Gauge Invariance ◽  
Cotangent Bundle ◽  
Symplectic Structure

scholarly journals On an inequality of Oprea for Lagrangian submanifolds

2009 ◽  
Vol 7 (1) ◽  
Author(s):  
Franki Dillen ◽  
Johan Fastenakels
Keyword(s):  
Complex Space ◽  
Space Form ◽  
Lagrangian Submanifolds ◽  
Lagrangian Submanifold ◽  
Complex Space Form ◽  
Totally Geodesic

AbstractWe show that a Lagrangian submanifold of a complex space form attaining equality in the inequality obtained by Oprea in [8], must be totally geodesic.

Lagrangian submanifolds satisfying a basic equality

1996 ◽  
Vol 120 (2) ◽  
pp. 291-307 ◽  
Author(s):  
Bang-Yen Chen ◽  
Luc Vrancken
Keyword(s):  
Isometric Immersion ◽  
Mean Curvature ◽  
Complex Space ◽  
Space Form ◽  
Lagrangian Submanifolds ◽  
Lagrangian Submanifold ◽  
Isometric Immersions ◽  
Equality Case ◽  
Basic Equality ◽  
Image Position

AbstractIn [3], B. Y. Chen proved that, for any Lagrangian submanifold M in a complex space-form Mn(4c) (c = ± 1), the squared mean curvature and the scalar curvature of M satisfy the following inequality:He then introduced three families of Riemannian n-manifolds and two exceptional n-spaces Fn, Ln and proved the existence of a Lagrangian isometric immersion pa from into ℂPn(4) and the existence of Lagrangian isometric immersions f, l, ca, da from Fn, Ln, , into ℂHn(− 4), respectively, which satisfy the equality case of the inequality. He also proved that, beside the totally geodesie ones, these are the only Lagrangian submanifolds in ℂPn(4) and in ℂHn(− 4) which satisfy this basic equality. In this article, we obtain the explicit expressions of these Lagrangian immersions. As an application, we obtain new Lagrangian immersions of the topological n-sphere into ℂPn(4) and ℂHn(−4).

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