Tangent sphere bundles satisfying \(\nabla_ \xi \tau=0\)
type: Item
instance of: scholarly article
title: Tangent sphere bundles satisfying \(\nabla_ \xi \tau=0\)
zbMATH Open document ID: 0794.53030
DOI: 10.1007/BF01228060
published in: Journal of Geometry
publication date: 1994-05-29T00:00:00Z
review text: Let \((N,G)\) be a Riemannian manifold and denote by \(T_ 1 N\) the unit tangent sphere bundle of it. Then \(T_ 1 N\) is endowed with the standard contact metric structure which we denote by \((\omega, g,\xi)\). Further, let \(\tau\) denote the Lie derivative of \(g\) with respect to the characteristic field \(\xi\). The main theorem of the paper states that \(\nabla_ \xi \tau=0\) on \((T_ 1 N,g)\) with respect to the Levi Civita connection \(\nabla\) if and only if \((N,g)\) has constant sectional curvature 0 or 1. The author also derives several equivalent conditions. They yield another proof of D. E. Blair's result: \((T_ 1 N,g)\) is locally symmetric if and only if \((N,G)\) is flat or two-dimensional and of constant sectional curvature 1.
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 558741
zbMATH Keywords: unit tangent sphere bundle
zbMATH Keywords: contact metric structure
zbMATH Keywords: constant sectional curvature
author: Domenico Perrone
reviewed by: Lieven Vanhecke
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 558741