On the geometry of the tangent bundle with the Cheeger-Gromoll metric
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 \(k \in (c_m, C_m]\).
scientific article; zbMATH DE number 1799653
scientific article