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

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.