Exponentially harmonic maps, Morse index and Liouville type theorems

We obtain a result on the Morse index of an exponentially harmonic map from a Riemannian manifold into the unit n-sphere. Next, we prove a Liouville type 1 theorem for exponentially harmonic maps between two Riemannian manifolds. Finally, let (M, g0) be a complete Riemannian manifold with a pole x0 and ( N, h) a Riemannian manifold, under certain conditions we establish a Liouville type 2 theorem for exponentially harmonic maps f : ( M,.2g0). N, 0 <.. C 8 (M).