  The first result in this study is a non-existence theorem for
$\alpha-$harmonic mappings. Additionally, a direct connection between the
$\alpha-$ harmonic and harmonic maps is made possible via conformal
deformation. Second, the instability of non-constant $\alpha$-harmonic maps is
investigated with regard to the target manifold's Ricci curvature requirements.
Next, the concept of $\alpha-$stable manifolds and their physical applications
are explored. Finally, it is investigated the $\alpha-$stability of compact
Riemannian manifolds that admit a non-isometric conformal vector field as well
as the Einstein Riemannian manifolds under certain assumption on the smallest
positive eigenvalue of its Laplacian operator on functions.
