The geometry of the Sasaki metric on the sphere bundle of Euclidean Atiyah vector bundles

Abstract

Let $(M,\\langle,\\rangle_{TM})$ be a Riemannian manifold. It is well-known that the Sasaki metric on $TM$ is very rigid but it has nice properties when restricted to $T^{(r)}M=\\{u\\in TM,|u|=r \\}$. In this paper, we consider a general situation where we replace $TM$ by a vector bundle $E\\longrightarrow M$ endowed with a Euclidean product $\\langle,\\rangle_E$ and a connection $\\nabla^E$ which preserves $\\langle,\\rangle_E$. We define the Sasaki metric on $E$ and we consider its restriction $h$ to $E^{(r)}=\\{a\\in E,\\langle a,a\\rangle_E=r^2 \\}$. We study the Riemannian geometry of $(E^{(r)},h)$ generalizing many results first obtained on $T^{(r)}M$ and establishing new ones. \nWe apply the results obtained in this general setting to the class of Euclidean Atiyah vector bundles introduced by the authors in arXiv preprint arXiv:1808.01254 (2018). Finally, we prove that any unimodular three dimensional Lie group $G$ carries a left invariant Riemannian metric such that $(T^{(1)}G,h)$ has a positive scalar curvature.