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

## TL;DR

This paper generalizes the geometry of the Sasaki metric from tangent bundles to Euclidean Atiyah vector bundles, analyzing their properties and applying findings to specific Lie groups with positive scalar curvature.

## Contribution

It introduces a generalized framework for the Sasaki metric on Euclidean vector bundles and explores its geometric properties, extending previous results on tangent bundles.

## Key findings

- The Sasaki metric on Euclidean Atiyah bundles exhibits specific geometric behaviors.
- Restrictions of the Sasaki metric to sphere bundles have notable properties.
- Certain Lie groups admit invariant metrics with positive scalar curvature on their tangent sphere 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 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.