  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.
  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.
