Caustics in geometrical optics
type: Item
instance of: scholarly article
title: Caustics in geometrical optics
zbMATH Open document ID: 0622.58004
DOI: 10.1007/BF01078474
author: Yu. V. Chekanov
publication date: 1986-01-01T00:00:00Z
review text: Let \(\pi\) : \(E\to B\) be a Lagrangian fibration. A hypersurface M in E is called layerwise convex if the intersection of M with each fiber is locally strictly convex in the fiber. For fixed M, a Lagrange submanifold L of E is called a ray manifold if L is contained in M. In geometrical optics it is natural to treat the ray manifolds, and in this paper a nontrivial restriction on the bifurcation of ''optical caustics'' for ray manifolds with respect to the projection \(\pi\) is given, comparing to the theory of Arnol'd on the classification of generic one-parameter bifurcations of caustics. Precisely, it is shown that if the singular locus \(\Sigma\) of a ray manifold with respect to \(\pi\) is nonsingular and compact, then the Euler characteristic of \(\Sigma\) is equal to zero, or equivalently, there exists a smooth direction field on \(\Sigma\). Some enumerative equalities are given also for the case \(\Sigma\) has a singularity. This paper is important for singularity theory and topology, and the reader is referred to \textit{V. I. Arnol'd}'s recent survey [Usp. Mat. Nauk 41, No.6, 3-18 (1986; Zbl 0618.58021)].
Mathematics Subject Classification ID: 58C25
Mathematics Subject Classification ID: 58K99
Mathematics Subject Classification ID: 78A05
Mathematics Subject Classification ID: 37J99
zbMATH DE Number: 4009362
zbMATH Keywords: layerwise convex hypersurface
zbMATH Keywords: Lagrangian singularity
zbMATH Keywords: Lagrangian fibration
zbMATH Keywords: ray manifold
zbMATH Keywords: bifurcation of ''optical caustics''
reviewed by: Goo Ishikawa
MaRDI profile type: Publication
cites work: Caustics and non-degenerate Hamiltonians
cites work: Wave front evolution and equivariant Morse lemma
full work available at URL: https://doi.org/10.1007/bf01078474
OpenAlex ID: W2086261276
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description: scientific article; zbMATH DE number 4009362