  Though Trudinger-Moser inequalities on compact Riemannian manifolds or
Euclidean space are well understood, we know little about them on complete
noncompact Riemannian manifolds. In this paper, we established respectively
necessary condition and sufficient condition under which Trudinger-Moser
inequalities hold on complete noncompact Riemannian manifolds. For applications
of such inequalities, we obtained existence results for some quasilinear
equations with nonlinearity of exponential growth.
