On infinitesimal \(C_{2\pi}\)-deformations of standard metrics on spheres
type: Item
instance of: scholarly article
title: On infinitesimal \(C_{2\pi}\)-deformations of standard metrics on spheres
zbMATH Open document ID: 0539.53032
author: Kazuyoshi Kiyohara
published in: Hokkaido Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: Let \(g_ t\) for any \(t\) be a Riemannian metric on \(S^ n\) whose geodesics are all closed and of length \(2\pi\) and where \(g_ 0\) is the standard metric. Then the symmetric 2-form \(h\) which is the derivative of \(g_ t\) with respect to \(t\) at \(t=0\) is called an infinitesimal \(C_{2\pi}\)-deformation of \(g_ 0\). \textit{V. Guillemin} [Adv. Math. 22, 85--119 (1976; Zbl 0353.53027)] proved that any symmetric 2-form \(h\) on \(S^ 2\) such that the integral of \(h({\dot \gamma},{\dot \gamma})\) along a great circle \(\gamma\) vanishes is a \(C_{2\pi}\)-deformation of \(g_ 0\). In the paper under review a necessary condition is introduced on 2-forms of the type \(fg_ 0\) to be an infinitesimal \(C_{2\pi}\)-deformation of \(g_ 0\), where \(f\) is a restriction of an odd polynomial on \({\mathbb R}^{n+1}\) to \(S^ n\). As a consequence it follows that Guillemin's result cannot be generalized to \(S^ n\) for \(n\geq 3\) without introducing new conditions on \(h\).
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 3857904
zbMATH Keywords: closed geodesics
zbMATH Keywords: \(C_{2\pi}\)-deformation
reviewed by: Gudlaugur Thorbergsson
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1381757715
OpenAlex ID: W2065841801
DOI: 10.14492/HOKMJ/1381757715
description: scientific article; zbMATH DE number 3857904