The Geometry of the Sasaki Metric on the Sphere Bundles 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 ∈ T M , | u | = 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 ∇ E which preserves ⟨ , ⟩ E . We define the Sasaki metric on E and we consider its restriction h to E (r) = { a ∈ E , ⟨ a , a ⟩ 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. [ABSTRACT FROM AUTHOR]