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

Abstract

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.