  In the first part of the paper we investigate some geometric features of
Moser-Trudinger inequalities on complete non-compact Riemannian manifolds. By
exploring rearrangement arguments, isoperimetric estimates, and gluing local
uniform estimates via Gromov's covering lemma, we provide a Coulhon,
Saloff-Coste and Varopoulos type characterization concerning the validity of
Moser-Trudinger inequalities on complete non-compact $n-$dimensional Riemannian
manifolds $(n\geq 2)$ with Ricci curvature bounded from below. Some sharp
consequences are also presented both for non-negatively and non-positively
curved Riemannian manifolds, respectively. In the second part, by combining
variational arguments and a Lions type symmetrization-compactness principle, we
guarantee the existence of a non-zero isometry-invariant solution for an
elliptic problem involving the $n-$Laplace-Beltrami operator and a critical
nonlinearity on $n-$dimensional homogeneous Hadamard manifolds. Our results
complement in several directions those of Y. Yang [J. Funct. Anal., 2012].
