On the effect of the domain on the number of orthogonal geodesic chords
type: Item
instance of: scholarly article
title: On the effect of the domain on the number of orthogonal geodesic chords
zbMATH Open document ID: 0903.53028
DOI: 10.1016/S0926-2245(96)00055-1
author: Fabio Giannoni
author: Pietro Majer
published in: Differential Geometry and its Applications
publication date: 1998-06-23T00:00:00Z
review text: A celebrated theorem of Lyusternik and Schnirelmann states that a convex subset of \({\mathbb{R}}^n\) with smooth boundary has at least \(n\) orthogonal geodesic chords. The purpose of this paper is to investigate how the number of orthogonal geodesic chords changes according to the topology of the domain. More precisely, let \(M\) be an \(n\)-dimensional (\(n\geq 2\)) Riemannian manifold of class \(C^3\) and let \(\overline{\Omega}\subset M\) be the closure of an open subset \(\Omega\). Assume that \(\partial\Omega\) is a manifold of class \(C^2\) and \(\overline\Omega\) is convex. There are two possibilities, according to the case where \(\overline{\Omega}\) has one or at least two holes: (i) \(\overline{\Omega}\) is homeomorphic to an annulus in \({\mathbb{R}}^n\). Then \(\overline{\Omega}\) has at least two geometrically distinct orthogonal geodesic chords. (ii) There exist \(p\geq 2\), \(x_1,\dots, x_p\in{\mathbb{R}}^n\), and positive numbers \(\rho_1,\dots, \rho_p\) such that \(\overline{B(x_i,\rho_i)}\subset B(0,1)\) for any \(i\), \(\overline{B(x_i,\rho_i)}\cap\overline{B(x_j,\rho_j)}=\emptyset\) if \(i\not=j\) and \(\overline{\Omega}\) is homeomorphic to \(\overline{B(0,1)}\setminus \bigcup_{i=1}^p \overline{B(x_i,\rho_i)}\). Then there exists a sequence \(\{\gamma_m\}_m\) of orthogonal geodesic chords in \(\overline{\Omega}\) having energy going to \(+\infty\). The problem of finding orthogonal geodesic chords is related to the search of brake orbits in a potential well, which represent a special kind of periodic solutions of a Hamiltonian system. In this direction the main results of this paper and that of Lyusternik and Schnirelmann suggest the possibility to have at least two brake orbits, provided that the potential well satisfies the above condition (i), and infinitely many whenever it satisfies (ii). The paper is well written and the multiplicity results obtained by the authors show that they depend on the topology of the manifold in a strange way.
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 58E10
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 1145945
zbMATH Keywords: Ljusternik-Schnirelmann theory
zbMATH Keywords: geodesics
zbMATH Keywords: multiplicity results
Wikidata QID: Q115337572
reviewed by: Vicenţiu D. Rădulescu
MaRDI profile type: Publication
cites work: Topological methods for variational problems with symmetries
cites work: Normal modes of a Lagrangian system constrained in a potential well
cites work: On the existence of closed geodesics on noncompact Riemannian manifolds
cites work: Q4146134
cites work: Kritische Sehnen auf Riemannschen Elementarraumstücken
cites work: Category of loop spaces of open subsets in euclidean space
cites work: Q4865364
cites work: Multiplicity of principal bounce trajectories with prescribed minimal period on Riemannian manifolds
cites work: On the existence of geodesics on stationary Lorentz manifolds with convex boundary
cites work: Morse relations for geodesics on stationary Lorentzian manifolds with boundary
cites work: Q3336423
cites work: Q3963718
cites work: Suspension Theorems and the Generalized Hopf Invariant
cites work: The imbedding problem for Riemannian manifolds
cites work: Morse theory on Hilbert manifolds
cites work: Homotopy theory of infinite dimensional manifolds
cites work: Periodic Solutions of Nonlinear Hyperbolie Partial Differential Equations
cites work: Q5601742
cites work: Periodische Bewegungen mechanischer Systeme
cites work: Closed geodesics on non-compact Riemannian manifolds
cites work: Q5530411
description: scientific article; zbMATH DE number 1145945