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

Let $(M,⟨,{⟩}_{TM})$ 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 $T^{(r)}M=\{u\in TM,\mathrm{\mid }u\mathrm{\mid }=r\}$. In this paper, we consider a general situation where we replace $TM$ by a vector bundle $E⟶M$ endowed with a Euclidean product $⟨,{⟩}_E$ and a connection ${\mathrm{\nabla }}^E$ which preserves $⟨,{⟩}_E$. We define the Sasaki metric on $E$ and we consider its restriction $h$ to $E^{®}={a\in E,\langle a,a\rangle_E=r^2 }$\mathrm{.}WestudytheRiemanniangeometryof$(E^{®},h)$generalizingmanyresultsfirstobtainedon$T^{®}M$ and establishing new ones. We 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.