ON THE GEOMETRY OF VECTOR BUNDLES WITH FLAT CONNECTIONS

Let E → M be an arbitrary vector bundle of rank k over a Riemannian manifold M equipped with a fiber metric and a compatible connection DE . R. Albuquerque constructed a general class of (twoweights) spherically symmetric metrics on E. In this paper, we give a characterization of locally symmetric spherically symmetric metrics on E in the case when DE is flat. We study also the Einstein property on E proving, among other results, that if k ≥ 2 and the base manifold is Einstein with positive constant scalar curvature, then there is a 1parameter family of Einstein spherically symmetric metrics on E, which are not Ricci-flat. Introduction and main results In the framework of Riemannian geometry, many special kinds of vector bundles have been considered and extensively studied, such as the cotangent bundle or the tangent bundle the literature of whose is very rich. Indeed, a wide range of interesting works have been published on the geometry of tangent bundles endowed with special types of metrics (Sasaki, Cheeger-Gromoll, . . . ) or more generally with g-natural metrics (cf. [1–3], [7]). For the general case of an arbitrary vector bundle, to the best of our knowledge, the situation becomes substantially different (cf. [5], [6]). Let (E, π,M) be a vector bundle equipped with a fiber metric h and a connection D compatible with h. Classically, the total space E, as a Riemannian manifold, have been “naturally” equipped with the metric π∗g ⊕ πh. More recently, in [4], R. Albuquerque considered a more general class of two-weights metrics with the weight functions depending on the fibre norm of E, i.e., metrics of the form g̃ = e2φ1π∗g ⊕ e2πh, where φ1, φ2 are smooth scalar functions on E depending only of the norm r = h(e, e) for e ∈ E, and smooth at r = 0 on the right. He called such metrics Received October 16, 2018; Revised March 1, 2019; Accepted March 8, 2019. 2010 Mathematics Subject Classification. 53C07, 53C24, 53C25.