A class of \(U(n)\)-invariant Riemannian metrics on manifolds diffeomorphic to odd-dimensional spheres
type: Item
instance of: scholarly article
title: A class of \(U(n)\)-invariant Riemannian metrics on manifolds diffeomorphic to odd-dimensional spheres
zbMATH Open document ID: 0815.53057
DOI: 10.1007/BF00975161
author: D. E. Vol'per
author: V. N. Berestovskij
published in: Siberian Mathematical Journal
publication date: 1994-11-08T00:00:00Z
review text: The authors study \(U(n)\)-invariant Riemannian metrics on the sphere \(S^{2n - 1} = U(n) / U(n - 1)\). Such metrics \(g\) up to a homothety depend on one parameter. The sectional curvature of \(g\) is calculated and the radius of injectivity is estimated. The authors show that some of these invariant metrics \(g\) on an odd dimensional sphere have a positive sectional curvature \(K\), \(0 < K \leq \lambda\), but the radius of injectivity \(d_ p\) in any point \(p\) does not satisfy Klingenberg's inequality \(d_ p \geq \pi/\sqrt{\lambda}\) which is valid for positively curved even dimensional manifolds. It is proved also that the sphere \(S^ 3\) with the standard metric of sectional curvature 1 is locally isometric to the sphere bundle of tangent vectors of length 1/2 on the 2- sphere \(S^ 2\) of radius 1/2 equipped with the Sasakian metric.
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C30
zbMATH DE Number: 652168
zbMATH Keywords: invariant Riemannian metrics
zbMATH Keywords: sectional curvature
zbMATH Keywords: radius of injectivity
Wikidata QID: Q115394646
reviewed by: Dmitri Alekseevsky
MaRDI profile type: Publication
cites work: A class of Riemannian homogeneous spaces
cites work: Q4746404
cites work: The fundamental equations of a submersion
cites work: On the Geometry of the Tangent Bundle.
description: scientific article; zbMATH DE number 652168