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J.Eells and L. Lemaire introduced $k$-harmonic maps, and Wang Shaobo showedthe first variation formula. In this paper, we give the second variationformula 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 sectionalcurvature, and show their non-trivial solutions.

The second variational formula of the k-energy and k-harmonic curves