Contact manifolds, Lagrangian Grassmannians and PDEs

Eshkobilov, Olimjon; Manno, Gianni; Moreno, Giovanni; Sagerschnig, Katja

doi:10.1515/coma-2018-0003

Cited by 3 publications

(10 citation statements)

References 33 publications

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“…28 The reader unfamiliar with EDS may take benefit from reading McKay's gentle introduction [56]. The peculiar relationship between contact manifolds and second order nonlinear PDEs is the main subject of two recent reviews [32,27]. 29 All relevant facts about LGr(n, 2n) are reviewed in [32].…”

Section: Second Order Pdes Associated To a Contact Cone Structurementioning

confidence: 99%

Complex contact manifolds, varieties of minimal rational tangents, and exterior differential systems

Buczyński

Moreno

2019

Banach Center Publ.

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Complex contact manifolds arise naturally in differential geometry, algebraic geometry and exterior differential systems. Their classification would answer an important question about holonomy groups. The geometry of such manifold X is governed by the contact lines contained in X. These are related to the notion of a variety of minimal rational tangents. In this review we discuss the partial classification theorems of projective complex contact manifolds. Among such manifolds one finds contact Fano manifolds (which include adjoint varieties) and projectivised cotangent bundles. In the first case we also discuss a distinguished contact cone structure, arising as the variety of minimal rational tangents. We discuss the repercussion of the aforementioned classification theorems for the geometry of quaternion-Kähler manifolds with positive scalar curvature and for the geometry of second-order PDEs imposed on hypersurfaces.Key words and phrases. contact geometry, projective complex contact manifolds, quaternion-Kähler manifolds, varieties of minimal rational tangents, exterior differential systems.1 Honestly, H is not a field, but this will not lead to any confusion. 2 The real-differentiable counterpart of this property would be the fact the points of a manifold are told apart by smooth functions: given two points there is always a smooth function taking different values on them, and given two tangent directions at a fixed point, there are two smooth functions with different corresponding partial derivatives.3 Strictly speaking, ampleness and positivity coincide whenever considering line bundles on complex projective manifolds.

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Section: Second Order Pdes Associated To a Contact Cone Structurementioning

confidence: 99%

Complex contact manifolds, varieties of minimal rational tangents, and exterior differential systems

Buczyński

Moreno

2019

Banach Center Publ.

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“…Besides countless scientific and technological applications, the problem of optimal mass transportation can be formulated in important economical models, in the form of optimal allocation of resources. This led, among many other things, to a Nobel prize in the economic sciences for Kantorovich [25,26,1,18]. On a more speculative level, one can ask for which functions f in (105) one obtains a (Cont(M )-invariant) class of PDEs.…”

Section: 5mentioning

confidence: 99%

“…[2, Formula (0.5)]. Bearing in mind the definition of Plücker coordinates (see (18), (20) and (21)), it is easy to see the geometry behind formula (110): it is nothing but the equation of a hyperplane section of LGr(n, 2n), namely the intersection of LGr(n, 2n) with the hyperplane (111)…”

Section: 5mentioning

confidence: 99%

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Geometry of Lagrangian Grassmannians and nonlinear PDEs

Gutt¹,

Manno

Moreno

2019

Banach Center Publ.

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This paper contains a thorough introduction to the basic geometric properties of the manifold of Lagrangian subspaces of a linear symplectic space, known as the Lagrangian Grassmannian. It also reviews the important relationship between hypersurfaces in the Lagrangian Grassmannian and second-order PDEs. Equipped with a comprehensive bibliography, this paper has been especially designed as an opening contribution for the proceedings volume of the homonymous workshop held in Warsaw, September 5-9, 2016, and organised by the authors.

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“…Of course the above definitions and reasonings can be localized in a neighborhood of a considered point. The classical literature about geometry of PDEs and their characteristics comprises, among others, [5,33]; see also the recent reviews [17,37].…”

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confidence: 99%

The moment map on the space of symplectic 3D Monge-Ampère equations

Gutt¹,

Manno²,

Moreno³

et al. 2021

Preprint

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For any 2 nd order scalar PDE E in one unknown function, that we interpret as a hypersurface of a secondorder jet space J 2 , we construct, by means of the characteristics of E, a sub-bundle of the contact distribution of the underlying contact manifold J 1 , consisting of conic varieties. We call it the contact cone structure associated with E. We then focus on symplectic Monge-Ampère equations in 3 independent variables, that are naturally parametrized by a 13-dimensional real projective space. If we pass to the field of complex numbers C, this projective space turns out to be the projectivization of the 14-dimensional irreducible representation of the simple Lie group Sp(6, C): the associated moment map allows to define a rational map from the space of symplectic 3D Monge-Ampère equations to the projectivization of the space of quadratic forms on a 6-dimensional symplectic vector space. We study in details the relationship between the zero locus of the image of , herewith called the cocharacteristic variety, and the contact cone structure of a 3D Monge-Ampère equation E: under the hypothesis of non-degenerate symbol, we prove that these two constructions coincide. A key tool in achieving such a result will be a complete list of mutually non-equivalent quadratic forms on a 6-dimensional symplectic space, which has an interest on its own. Contents 1. Introduction 1 2. Preliminaries 3 3. The contact cone structure associated with a second-order PDE 7 4. Quadric contact cone structures associated with 3D Monge-Ampère equations 10 5. Reconstructing a second-order PDE from a contact cone structure 13 6. The space PΛ 3 0 (C) of symplectic 3D Monge-Ampère equations 15 7. Normal forms of quadratic forms on C with respect to Sp(C) 18 8. The moment map on the space of Monge-Ampère equations 25 9. Hyperplane sections of the Lagrangian Grassmannian LGr(3, C) 30 10. Conclusions and perspectives 35 References 35

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Contact manifolds, Lagrangian Grassmannians and PDEs

Cited by 3 publications

References 33 publications

Complex contact manifolds, varieties of minimal rational tangents, and exterior differential systems

Complex contact manifolds, varieties of minimal rational tangents, and exterior differential systems

Geometry of Lagrangian Grassmannians and nonlinear PDEs

The moment map on the space of symplectic 3D Monge-Ampère equations

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