On the geometry of the tangent bundle with the Cheeger-Gromoll metric
type: Item
instance of: scholarly article
title: On the geometry of the tangent bundle with the Cheeger-Gromoll metric
zbMATH Open document ID: 1019.53017
author: Sigmundur Gudmundsson
author: Elias Kappos
published in: Tokyo Journal of Mathematics
publication date: 2002-11-05T00:00:00Z
review text: Let \((M, g)\) be an \(m\)-dimensional Riemannian manifold of constant sectional curvature \(k\). In the present paper the authors study the tangent bundle \(TM\) of \((M, g)\) endowed with the Cheeger-Gromoll metric \(\tilde g\) induced by \(g\). In [\textit{M. Sekizawa}, Tokyo J. Math. 14, 407-417 (1991; Zbl 0768.53020)] it was stated that if \(k \geq - 3 (m - 2) / m\), then \((TM, \tilde g)\) has non-negative scalar curvature. The authors give counterexamples to the above statement. More precisely, they prove that if \(m = 2\) then there exists a real number \(C_2 \geq 40\) such that \((TM, {\tilde g})\) has positive scalar curvature if and only if \(k \in [0, C_2)\) and it has non-negative scalar curvature if and only if \(k \in [0, C_2]\). In higher dimension \(m > 2\), they show that there exist real numbers \(c_m < 0\) and \(C_m > 60\) such that \((TM, \tilde g)\) has positive scalar curvature if and only if \(k \in (c_m, C_m)\) and it has non-negative scalar curvature if and only if \(k \in (c_m, C_m]\).
reviewed by: Anna Fino
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 1799653
zbMATH Keywords: tangent bundles
zbMATH Keywords: natural metrics
zbMATH Keywords: positive scalar curvature
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1244208938
OpenAlex ID: W2003849290
cites work: On the structure of complete manifolds of nonnegative curvature
cites work: On the Geometry of the Tangent Bundle.
cites work: On the geometry of harmonic morphisms
cites work: Riemannian metrics on tangent bundles
cites work: On the differential geometry of tangent bundles of Riemannian manifolds
cites work: Curvatures of tangent bundles with Cheeger-Gromoll metric
DOI: 10.3836/TJM/1244208938
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description: scientific article; zbMATH DE number 1799653