The geometry of generalized Cheeger-Gromoll metrics on the total space of transitive Euclidean Lie algebroids

TL;DR

This paper explores the geometric properties of generalized Cheeger-Gromoll metrics on the total space of transitive Euclidean Lie algebroids, extending known results from tangent bundles and revealing new curvature phenomena.

Contribution

It introduces natural metrics on transitive Euclidean Lie algebroids, proves a rigidity result for these metrics, and analyzes their properties on Atiyah Lie algebroids, especially over space forms.

Findings

Generalized Cheeger-Gromoll metrics can be naturally extended to transitive Euclidean Lie algebroids.

A new rigidity theorem generalizes previous results for tangent bundles.

Conditions for positive scalar curvature on Atiyah Euclidean Lie algebroids over space forms.

Abstract

Natural metrics (Sasaki metric, Cheeger-Gromoll metric, Kaluza-Klein metrics etc.. ) on the tangent bundle of a Riemannian manifold is a central topic in Riemannian geometry. Generalized Cheeger-Gromoll metrics is a family of natural metrics $h_{p,q}$ depending on two parameters with $p\in\mathbb{R}$ and $q\geq0$. This family has been introduced recently and possesses interesting geometric properties. If $p=q=0$ we recover the Sasaki metric and when $p=q=1$ we recover the classical Cheeger-Gromoll metric. A transitive Euclidean Lie algebroid is a transitive Lie algebroid with an Euclidean product on its total space. In this paper, we show that natural metrics can be built in a natural way on the total space of transitive Euclidean Lie algebroids. Then we study the properties of generalized Cheeger-Gromoll metrics on this new context. We show a rigidity result of this metrics which generalizes so far all rigidity results known in the case of the tangent bundle. We show also that considering natural metrics on the total space of transitive Euclidean Lie algebroids opens new interesting horizons. For instance, Atiyah Lie algebroids constitute an important class of transitive Lie algebroids and we will show that natural metrics on the total space of Atiyah Euclidean Lie algebroids have interesting properties. In particular, if $M$ is a Riemannian manifold of dimension $n$, then the Atiyah Lie algebroid associated to the $\mathrm{O}(n)$-principal bundle of orthonormal frames over $M$ possesses a family depending on a parameter $k>0$ of transitive Euclidean Lie algebroids structures say $AO(M,k)$. When $M$ is a space form of constant curvature $c$, we show that there exists two constants $C_n<0$ and $K(n,c)>0$ such that $(AO(M,k),h_{1,1})$ is a Riemannian manifold with positive scalar curvature if and only if $c>C_n$ and $0<k\leq K(n,c)$.