  J.Eells and L. Lemaire introduced $k$-harmonic maps, and Wang Shaobo showed
the first variation formula. In this paper, we give the second variation
formula of $k$-energy, and give a notion of index, nullity and weakly stable.
We also study $k$-harmonic maps into the product Riemannian manifold, and
$k$-harmonic curves into a Riemannian manifold with constant sectional
curvature, and show their non-trivial solutions.
