Generalized billiard paths and Morse theory for manifolds with corners
type: Item
instance of: scholarly article
title: Generalized billiard paths and Morse theory for manifolds with corners
zbMATH Open document ID: 1018.57015
DOI: 10.1016/S0166-8641(02)00036-6
author: David G. C. Handron
published in: Topology and its Applications
publication date: 2003-04-02T00:00:00Z
review text: Let \(M\hookrightarrow \mathbb{R}^N\) be a smooth \(n\)-manifold embedded in Euclidean space dimension \(N\). If \(p,q\in \mathbb{R}^N\), then a generalized billiard path with \(k\)-reflections connecting \(p\) to \(q\) is an ordered sequence \(P=\{\alpha_1, \dots,\alpha_k\}\) of points \(\alpha_1,\dots,\alpha_k\in M\) such that for each \(i=1,\dots,n\), one of the following is true: (1) The bisector of \(\sphericalangle \alpha_{i-1}\alpha_i \alpha_{i+1}\) is normal to \(T_{\alpha_i} M\). (2) \(\sphericalangle \alpha_{i-1} \alpha_i \alpha_{i+1}\) is a straight angle where \(\alpha_0=p\) and \(\alpha_{k+1}=q\). This definition allows the line segment \(\overline{\alpha_i \alpha_{i+1}}\) to intersect the manifold. If \(M\) happens to be a convex hypersurface, the definition reduces to the usual notion of billiard path. The author desires to estimate the number of generalized billiard paths connecting fixed points with a given number of reflections. The length of a series of line segments with \(k\) reflections may be thought of as a function on \(M^k\) and generalized billiard paths correspond to critical points of this length function. The length function is not smooth on \(M^k\), having singularities along its diagonals \(\{\alpha_i= \alpha_{i+1}\}\). But following the Fulton-MacPherson procedure [Ann. of Math. 139, 183-225 (1994; Zbl 0820.14037)] one can `blow up' \(M^k\) to obtain a compact manifold with corners \(X_k\) to which the length function extends smoothly. In this situation the author develops a version of Morse theory for manifolds with corners and uses it to study this length function. This Morse theory is a generalization of that of \textit{D. Braess} [Math. Ann. 208, 133-148 (1974; Zbl 0263.58005)], retaining both a global `gradient' flow and the intrinsic stratification of a manifold with corners. Then the author finds that the number of generalized billiard path with \(k\) reflection connecting two points in \(\mathbb{R}^N\) can be estimated in ter…
reviewed by: Ioan Pop
Mathematics Subject Classification ID: 57R70
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 57R25
zbMATH DE Number: 1891201
zbMATH Keywords: Morse theory
zbMATH Keywords: Stratified spaces
zbMATH Keywords: Billiard path problem
MaRDI profile type: Publication
cites work: Morse theory indomitable
cites work: Morse Theorie für berandete Mannigfaltigkeiten. (Morse theory for bounded manifolds)
cites work: A compactification of configuration spaces
cites work: Q4039743
cites work: On stratified Morse theory
cites work: Morse lemmas for smooth functions on manifolds with corners
Wikidata QID: Q126011192
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description: scientific article; zbMATH DE number 1891201