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Let $(M,\langle,\rangle_{TM})$ be a Riemannian manifold. It is well-knownthat the Sasaki metric on $TM$ is very rigid but it has nice properties whenrestricted to $T^{(r)}M=\{u\in TM,|u|=r \}$. In this paper, we consider ageneral 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 $E^{(r)}=\{a\in E,\langlea,a\rangle_E=r^2 \}$. We study the Riemannian geometry of $(E^{(r)},h)$generalizing many results first obtained on $T^{(r)}M$ and establishing newones.  We apply the results obtained in this general setting to the class ofEuclidean Atiyah vector bundles introduced by the authors in arXiv preprintarXiv:1808.01254 (2018). Finally, we prove that any unimodular threedimensional Lie group $G$ carries a left invariant Riemannian metric such that$(T^{(1)}G,h)$ has a positive scalar curvature.

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