Homogeneous Riemannian almost P-manifolds
type: Item
instance of: scholarly article
title: Homogeneous Riemannian almost P-manifolds
zbMATH Open document ID: 0726.53033
DOI: 10.1007/BF00974492
author: E. D. Rodionov
published in: Siberian Mathematical Journal
publication date: 1990-01-01T00:00:00Z
review text: A Riemannian homogeneous space is called a P-manifold if all its geodesics have a positive common period, and an almost P-manifold if all proper totally geodesic submanifolds of dimension \(\geq 2\) are isometric to a P-manifold. The following theorem is proved. Let \(G\) be a compact Lie group and \(M=G/H\) a simply connected P-manifold which is not isometric to a compact Lie group endowed with a left-invariant metric. If \(\dim M\) is even then \(M\) is isometric to a symmetric space of rank 1, while for odd \(\dim M\) we have \(\operatorname{rk}G=\operatorname{rk}H+1\). The proof makes use of the following fact: for any homogeneous Riemannian manifold \(G/H\), the submanifold \(C(T)/C(T)\cap H\), where \(C(T)\) is the centralizer of a maximal torus \(T\) of \(H\), is totally geodesic in \(M\).
Mathematics Subject Classification ID: 53C30
zbMATH DE Number: 4198961
zbMATH Keywords: almost P-manifold
zbMATH Keywords: totally geodesic submanifolds
zbMATH Keywords: compact Lie group
zbMATH Keywords: symmetric space
Wikidata QID: Q115394651
MaRDI profile type: Publication
cites work: Q3283409
cites work: Compact homogeneous Riemannian manifolds with strictly positive curvature
cites work: Q3040034
cites work: Q3913422
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cites work: Les variétés riemanniennes homogenes simplement connexes de dimension impaire à courbure strictement positive
full work available at URL: https://doi.org/10.1007/bf00974492
OpenAlex ID: W2024542670
description: scientific article; zbMATH DE number 4198961