On the geometry of vector bundles with flat connections

Bull. Korean Math. Soc. Published online May 29, 2019

Mohamed Tahar Kadaoui Abbassi and Ibrahim Lakrini
University Sidi Mohammed Benabdallah, Faculty of Sciences Dhar El Mahraz

Abstract : Let $E \longrightarrow M$ be an arbitrary vector bundle over a Riemannian manifold $M$ equipped with a fiber metric and a compatible connection $D^{E}$. R. Albuquerque constructed a general class of (two-weights) spherically symmetric metrics on $E$. In this paper, we equip the manifold $E$ with a spherically symmetric metric and we prove that the properties of being flat, or of constant sectional curvature or of constant scalar curvature, or an Einstein manifold are hereditary from the base manifold. We then suppose that $D^{E}$ is flat, and we give necessary and sufficient conditions for $E$ to be locally symmetric. We prove also that if the base manifold is Einstein of positive constant scalar curvature, then there is a 1-parameter family of locally symmetric Einstein spherically symmetric metrics $E$, which are not Ricci-flat. We then show that $E$ can not be of constant sectional curvature unless it is flat and that in this case the base manifold is also flat. We give also some results concerning the sign of the scalar curvature of $E$. Finally we prove that if the sectional curvature of $E$ is bounded then that of $M$ is also bounded, and we give some situations when the converse is true.

Keywords : Vector bundle, spherically symmetric metric, curvatures, Einstein manifold, local symmetry.