Curvatures of the tangent bundle with a special almost product metric
type: Item
instance of: scholarly article
title: Curvatures of the tangent bundle with a special almost product metric
zbMATH Open document ID: 1246.53023
DOI: 10.1134/S000143461103028X
author: O. V. Sukhova
published in: Mathematical Notes
publication date: 2011-11-25T00:00:00Z
review text: Let \((M,g)\) be a Riemannian manifold. The Levi-Civita connection of \(g\) gives the horizontal distribution \(H\) on the total space of the tangent bundle \(\pi : TM \to M\), and \(TM = H \oplus V\), where \(V\) is the vertical distribution on \(TM\). Each vector field \(X\) on \(M\) determines two vector fields on \(TM\): the horizontal lift \(X^h\) and the vertical lift \(X^v\). On \(TM\) the author considers a Riemannian metric \(\widetilde g\) given by the following properties: for \(X,Y \in H\), \[ \widetilde g_y(X^h,Y^h) = g_{\pi(y)}(X,Y), \quad \widetilde g_y(X^h,Y^v)=0, \] \[ g_y(X^v,Y^v) = \varphi(z) g_{\pi(y)}(X,Y) + \psi(z) g_{\pi(y)}(X,y)g_{\pi(y)}(Y,y), \] where \(z = z(y) = \frac{1}{2}g(y,y)\), \(\varphi\) and \(\psi\) are functions such that \(\varphi > 0\) and \(\varphi + 2z \psi > 0\). The main result of the paper is an expression for the sectional curvature of \((TM,\widetilde g)\) in terms of the curvature tensor of \((M,g)\) and the functions \(\varphi\) and \(\psi\), from which the author obtains some results about the scalar curvature of \(\widetilde g\) for metrics \(g\) of constant sectional curvature. For example, if \((M,g)\) has constant sectional curvature \(k\), then for \(\varphi=1\) and \(\psi = c = const > 0\), \((TM, \widetilde g)\) has negative scalar curvature if and only if \(k \in (-\infty,-c]\).
reviewed by: M. A. Malakhaltsev
Mathematics Subject Classification ID: 53B20
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 5980803
zbMATH Keywords: tangent bundle
zbMATH Keywords: almost product metric
zbMATH Keywords: Sasaki metric
zbMATH Keywords: Cheeger-Gromoll metric
zbMATH Keywords: scalar curvature
zbMATH Keywords: sectional curvature
MaRDI profile type: Publication
cites work: On the differential geometry of tangent bundles of Riemannian manifolds
cites work: Curvatures of tangent bundles with Cheeger-Gromoll metric
cites work: On the geometry of the tangent bundle with the Cheeger-Gromoll metric
cites work: Q5538378
full work available at URL: s000143461103028x
OpenAlex ID: W2017962434
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description: scientific article; zbMATH DE number 5980803