  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 $0<k\leq K(n,c)$.
