Curvatures of the tangent bundle with a special almost product metric
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 \(k \in (-\infty,-c]\).
scientific article; zbMATH DE number 5980803
scientific article