Conjugate points on geodesics of Hofer's metric
type: Item
instance of: scholarly article
title: Conjugate points on geodesics of Hofer's metric
zbMATH Open document ID: 0864.58004
DOI: 10.1016/S0926-2245(96)00027-7
author: Ilya Ustilovsky
published in: Differential Geometry and its Applications
publication date: 1997-05-26T00:00:00Z
review text: Hofer's metric on the group of Hamiltonian diffeomorphisms of a symplectic manifold is generated by the \(L^\infty\)-norm on the Lie algebra. In this paper we develop a variational theory of geodesics of this metric which satisfy certain non-degeneracy assumptions. We derive the second variation formula, describe conjugate points and obtain necessary and sufficient conditions for the \(C^\infty\)-local minimality of such geodesics. We also present an example of a nondegenerate geodesic which is not locally minimal at its first conjugate point.
Mathematics Subject Classification ID: 58D05
Mathematics Subject Classification ID: 58E10
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C60
zbMATH DE Number: 937462
zbMATH Keywords: Finslerian geometry
zbMATH Keywords: Hofer metric
zbMATH Keywords: Hamiltonian diffeomorphisms
zbMATH Keywords: symplectic manifold
zbMATH Keywords: second variation
MaRDI profile type: Publication
cites work: Q4692740
cites work: Geodesics of Hofer's metric on the group of Hamiltonian diffeomorphisms
cites work: Q4039749
cites work: On the topological properties of symplectic maps
cites work: Estimates for the energy of a symplectic map
cites work: Q4850937
cites work: The geometry of symplectic energy
cites work: Local non-squeezing theorems and stability
cites work: New minimal geodesics in the group of symplectic diffeomorphisms
Wikidata QID: Q115337653
description: scientific article; zbMATH DE number 937462