Displacing (Lagrangian) submanifolds in the manifolds of full flags

2015 ◽  
Vol 15 (1) ◽  
Author(s):  
Milena Pabiniak
Keyword(s):  
Symplectic Geometry ◽  
Lagrangian Submanifolds ◽  
Lagrangian Submanifold ◽  
Parameter Family ◽  
Coadjoint Orbits ◽  
Large Collection ◽  
Coadjoint Action

AbstractIn symplectic geometry a question of great importance is whether a (Lagrangian) submanifold is displaceable, that is, if it can be made disjoint from itself by a Hamiltonian isotopy.We analyze the coadjoint orbits of SU(n) and their Lagrangian submanifolds that are the fibers of the Gelfand-Tsetlin map.We use the coadjoint action to displace a large collection of these fibers. Thenwe concentrate on the case n = 3 and apply McDuff’s method of probes to show that “most” of the generic Gelfand-Tsetlin fibers are displaceable. “Most” means “all but one” in the non-monotone case, and it means “all but a 1-parameter family” in the monotone case. In the case of a non-monotone manifold of full flags we present explicitly a unique non-displaceable Lagrangian fiber (S

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 Quasi Semi-Border Singularities

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 495
Author(s):  
Fawaz Alharbi ◽  
Suliman Alsaeed
Keyword(s):  
Spectral Sequence ◽  
Equivalence Relation ◽  
Symplectic Geometry ◽  
Lagrangian Submanifolds ◽  
Natural Extension ◽  
Main Method ◽  
Equivalent Relation ◽  
Sequence Method ◽  
Artificial Nature

We obtain a list of simple classes of singularities of function germs with respect to the quasi m-boundary equivalence relation, with m ≥ 2 . The results obtained in this paper are a natural extension of Zakalyukin’s work on the new non-standard equivalent relation. In spite of the rather artificial nature of the definitions, the quasi relations have very natural applications in symplectic geometry. In particular, they are used to classify singularities of Lagrangian projections equipped with a submanifold. The main method that is used in the classification is the standard Moser’s homotopy technique. In addition, we adopt the version of Arnold’s spectral sequence method, which is described in Lemma 2. Our main results are Theorem 4 on the classification of simple quasi classes, and Theorem 5 on the classification of Lagrangian submanifolds with smooth varieties. The brief description of the main results is given in the next section.

scholarly journals Geometry of Thermodynamic Processes

Entropy ◽  
2018 ◽  
Vol 20 (12) ◽  
pp. 925 ◽  
Author(s):  
Arjan van der Schaft ◽  
Bernhard Maschke
Keyword(s):  
Symplectic Geometry ◽  
Hamiltonian Dynamics ◽  
Contact Geometry ◽  
Thermodynamic System ◽  
Lagrangian Submanifold ◽  
Geometric Framework ◽  
Definition Of ◽  
And Control ◽  
Geometric Formulation ◽  
Thermodynamic Processes

Since the 1970s, contact geometry has been recognized as an appropriate framework for the geometric formulation of thermodynamic systems, and in particular their state properties. More recently it has been shown how the symplectization of contact manifolds provides a new vantage point; enabling, among other things, to switch easily between the energy and entropy representations of a thermodynamic system. In the present paper, this is continued towards the global geometric definition of a degenerate Riemannian metric on the homogeneous Lagrangian submanifold describing the state properties, which is overarching the locally-defined metrics of Weinhold and Ruppeiner. Next, a geometric formulation is given of non-equilibrium thermodynamic processes, in terms of Hamiltonian dynamics defined by Hamiltonian functions that are homogeneous of degree one in the co-extensive variables and zero on the homogeneous Lagrangian submanifold. The correspondence between objects in contact geometry and their homogeneous counterparts in symplectic geometry, is extended to the definition of port-thermodynamic systems and the formulation of interconnection ports. The resulting geometric framework is illustrated on a number of simple examples, already indicating its potential for analysis and control.

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).

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 Geometry of G2 orbits and isoparametric hypersurfaces

2011 ◽  
Vol 203 ◽  
pp. 175-189
Author(s):  
Reiko Miyaoka
Keyword(s):  
Lie Algebra ◽  
Lagrangian Submanifolds ◽  
Point Of View ◽  
Parameter Family ◽  
Isoparametric Hypersurfaces ◽  
Singular Orbits

AbstractWe characterize the adjointG2orbits in the Lie algebragofG2as fibered spaces overS6with fibers given by the complex Cartan hypersurfaces. This combines the isoparametric hypersurfaces of case (g,m) = (6,2) with case (3,2). The fibrations on two singular orbits turn out to be diffeomorphic to the twistor fibrations ofS6andG2/SO(4). From the symplectic point of view, we show that there exists a 2-parameter family of Lagrangian submanifolds on every orbit.

Symplectic topology of Lagrangian submanifolds of ℂPn with intermediate minimal Maslov numbers

2017 ◽  
Vol 17 (2) ◽  
pp. 247-264 ◽  
Author(s):  
Hiroshi Iriyeh
Keyword(s):  
Lagrangian Submanifolds ◽  
First Integral ◽  
Lagrangian Submanifold ◽  
Integral Homology ◽  
Symplectic Topology ◽  
Floer Cohomology ◽  
Topological Features

AbstractWe examine symplectic topological features of a certain family of monotone Lagrangian submanifolds in ℂPn. First we give cohomological constraints on a Lagrangian submanifold in ℂPn whose first integral homology is p-torsion. In the case where (n, p) = (5,3), (8, 3), we prove that the cohomologies with coefficients in ℤ2 of such Lagrangian submanifolds are isomorphic to that of SU(3)/(SO(3)ℤ3) and SU(3)/ℤ3, respectively. Then we calculate the Floer cohomology with coefficients in ℤ2 of a monotone Lagrangian submanifold SU(p)/ℤp in ${\mathbb C}P^{p^2-1}.$

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