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

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 hp,q of natural metrics on the tangent bundle depending on two parameters with p∈R and q≥0. This family 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 a Euclidean product on its total space. In this paper, we show that the analogous of natural metrics 11Natural metrics on the tangent bundles come from a classification of natural transformation of Riemannian metrics on manifolds to metrics on tangent bundles. We use the some terminology in this paper to designate metrics obtained on the total space of a Euclidean Lie algebroid by imitating the expression of g-natural metrics on the tangent bundle.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 these metrics which generalizes some 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 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 exist two constants Cn<0 and K(n,c)>0 such that (AO(M,k),h1,1) is a Riemannian manifold with positive scalar curvature if and only if c>Cn and 0<k≤K(n,c).