The Geometry of the Sasaki Metric on the Sphere Bundles of Euclidean Atiyah Vector Bundles

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 Boucetta and Essoufi J Geom Phys 140:161–177, 2019). 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.