Curvatures of tangent bundles with Cheeger-Gromoll metric
The author studies the Levi-Civita connection, the Riemannian curvature and the scalar curvature of the Cheeger-Gromoll metric defined on the tangent bundle \(TM\) of an \(n\)-dimensional Riemannian manifold \(M\) [see, for the definition, \textit{J. Cheeger} and \textit{D. Gromoll}, Ann. Math., II. Ser. 96, 413-443 (1972; Zbl 0246.53049)]. The main result of the paper states that the scalar curvature of \(TM\) is non negative if the original metric of \(M\) has constant curvature \(c \geq -3(n-2)/n\), \(n = \dim M\). In particular, this implies that the scalar curvature at \((x,u) \in TM\) is never constant if the orignal metric on \(M\) has constant curvature.
scientific article; zbMATH DE number 60849