On the curvature of the induced Riemannian metric on the frame bundle of a Riemannian manifold
Let (M,G) be a Riemannian manifold, FM the frame bundle of M and \(G^ D\) the diagonal lift of G to FM with respect to the Levi-Civita connection of G [same authors, Rend. Circ. Mat. Palermo, II. Ser. 32, 236-271 (1983; Zbl 0524.53020)]; then \(G^ D\) actually is the Riemannian metric firstly considered by \textit{K. P. Mok} [J. Reine Angew. Math. 302, 16-31 (1978; Zbl 0378.53016)]. In this paper, the Levi-Civita connection of \(G^ D\) and its curvature tensor are computed, obtaining formulae for them similar to those given by \textit{O. Kowalski} [J. Reine Angew. Math. 250, 124-129 (1971; Zbl 0222.53044)] for the tangent bundle TM endowed with the Sasaki metric. Using these formulae the sectional, Ricci and scalar curvatures of \((FM,G^ D)\) are computed and related with the corresponding curvatures of (M,G); as a consequence some results on the differential geometry of \((FM,G^ D)\) are obtained, as for instance: Theorem: If \((FM,G^ D)\) is an Einstein manifold, then (M,G) is flat.
scientific article; zbMATH DE number 3861951