On Lagrangian and Legendrian Singularities
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On Lagrangian and Legendrian Singularities Vyacheslav D. Sedykh1 Received: 12 June 2018 / Revised: 19 April 2019 / Accepted: 11 September 2020 © Institute for Mathematical Sciences (IMS), Stony Brook University, NY 2020
Abstract We describe the topology of stable simple multisingularities of Lagrangian and Legendrian maps. In particular, the tables of adjacency indices of monosingularities to multisingularities are given for generic caustics and wave fronts in spaces of small dimensions. The paper is an extended version of the author’s talk in the International Conference “Contemporary mathematics” in honor of the 80th birthday of V. I. Arnold (Moscow, Russia, 2017). Keywords Lagrangian and Legendrian maps · Caustics · Wave fronts · (Multi)singularities · Adjacency index
1 Introduction ”Caustic” is a term from Optics. It refers to a place where light is concentrated. A caustic is the envelope of a system of light rays. For example, we can see a caustic on the bottom of a cup in a Sunny day (Fig. 1). A caustic is the set of critical values of so-called Lagrangian map (see [1]). A generic caustic is a singular hypersurface in the target space. Singularities of this hypersurface are defined by multisingularities of the corresponding Lagrangian map. In particular, a generic caustic in the plane may have only cusps and transversal intersections of two smooth branches. A simple example of a wave front is an equidistant of a smooth (of class C ∞ ) cooriented hypersurface in the Euclidean space Rn (Fig. 2). A wave front is the image of so-called Legendrian map [1]. A generic wave front is a singular hypersurface in the target space. Singularities of this hypersurface are defined by multisingularities of the corresponding Legendrian map. In particular, a generic wave front in the plane may have only cusps and transversal intersections of two smooth branches.
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Vyacheslav D. Sedykh [email protected] ; [email protected] Department of Higher Mathematics, National University of Oil and Gas “Gubkin University”, Leninsky prospect 65, Moscow 119991, Russia
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V. D. Sedykh
Fig. 1 Caustic on the bottom of a cup
Fig. 2 An equidistant of an ellipse in the plane
Stable simple singularities of caustics and wave fronts were classified by V. I. Arnold [1]. He repeatedly posed the natural problem on the local and global topological properties of caustics and wave fronts (see Problem 1987-5 in [2]). This problem was the subject of numerous works of different authors (see, for example, the book [11]). However, even local problem is not solved completely until now. In this paper we describe the recent results on the topology of adjacencies of stable simple multisingularities of Lagrangian and Legendrian maps. Adjacency indices of monosingularities to multisingularities are given for generic caustics in the spaces of dimension n ≤ 5 and for generic wave fronts in the spaces of dimension n ≤ 6.
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On Lagrangian and Legendrian Singularities
2 Lagrangian Singularities Let E be a smooth manifold. A symplectic structure on E is a c
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