A curvature characterization of certain locally rank-one symmetric spaces
type: Item
instance of: scholarly article
title: A curvature characterization of certain locally rank-one symmetric spaces
zbMATH Open document ID: 0654.53053
author: Quo-Shin Chi
published in: Journal of Differential Geometry
publication date: 1988-01-01T00:00:00Z
review text: Let R(X,Y)Z denote the curvature tensor of a Riemannian manifold M. Define the Jacobi operator \(K_ v(.):=R(.,v)v\) for each unit tangent vector v. Suppose \(K_ v\) has constant eigenvalues (counting multiplicities) independent of v. It is proved that in this case 1. if M is odd dimensional, then M is of constant sectional curvature, 2. if dim M\(=2(2k+1)\), \(k\geq 0\), or dim M\(=4\), then M has either constant curvature or is covered by a standard complex projective space or its noncompact dual, 3. if M is nonflat and Kählerian, then M has constant holomorphic sectional curvature if the sectional curvatures are all nonpositive or all nonnegative.
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 4067804
zbMATH Keywords: locally rank-one symmetric space
zbMATH Keywords: Kähler manifold
zbMATH Keywords: constant curvature
zbMATH Keywords: sectional curvatures
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214442277
OpenAlex ID: W1554474935
Wikidata QID: Q115183542
DOI: 10.4310/JDG/1214442277
description: scientific article; zbMATH DE number 4067804