Flow-symmetric Riemannian manifolds
type: Item
instance of: scholarly article
title: Flow-symmetric Riemannian manifolds
zbMATH Open document ID: 0957.53016
author: Peter Bueken
author: Lieven Vanhecke
author: Eric Boeckx
published in: Beiträge zur Algebra und Geometrie
publication date: 2000-06-18T00:00:00Z
full work available at URL: https://eudml.org/doc/227264
full work available at URL: 14.html
review text: The authors introduce the notion of locally flow-symmetric Riemannian manifolds as Riemannian manifolds admitting a non-vanishing vector field \(\xi\) such that the local reflections with respect to the integral curves of this vector field are local isometries. They give the necessary conditions (on the Riemann curvature tensor and its covariant derivatives) for a Riemannian manifold to be locally flow-symmetric and show that these conditions are also sufficient in the analytic case. The authors find that the manifold can only be locally flow-symmetric with respect to the eigenvector fields of the Ricci operator. They observe that all locally \(\phi\)-symmetric Sasakian manifolds, strongly \(\phi\)-symmetric contact metric manifolds and locally KTS-spaces are examples of locally flow-symmetric manifolds, and construct a number of new examples. The authors also prove that the warped product manifold \(\mathbb{R}\times_f M\) is locally flow-symmetric if and only if \(M\) is a locally symmetric manifold. Finally, using this fact, they present a complete local classification of two-dimensional locally flow-symmetric spaces.
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 1460290
zbMATH Keywords: locally flow-symmetric space
zbMATH Keywords: KTS-space
zbMATH Keywords: \(\phi\)-symmetric Sasakian space
zbMATH Keywords: strongly \(\phi\)-symmetric contact metric manifold
reviewed by: Marian Hotloś
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1460290