The Relationship Between K-harmonic Maps And Harmonic Maps

In this paper, we mainly study the k-harmonic map ? (k?2) from complete Riemannian manifold (M, g) into Riemannian manifold (N, h) with non-positive curvature. The energy and j-energy of ? are denoted as E1(?)= E(?) and Ej(?). We get the following result which can generalize the relevant conclusion for 2-harmonic maps.(i) Every k-harmonic map ?:(M, g)?(N, h) (k?2) with finite j-energy until j= 2k-2, must be harmonic.(ii) In the case of Vol(M,g)=?, every k-harmonic map ?:(M, g)? (N,h) (k?2) with finite j-energy Ej(?) for all j= 2,4,…,2k-2 is harmonic.In addition, as the application of a k-harmonic map, a submanifold (M,g) is called a k-harmonic submanifold in (N, h) if the map ?:(M, g)?(N, h)(k? 2) is an isometric immersion. Therefore we consider the 3-harmonic submanifolds especially, and obtain sonic sufficient conditions of the 3-harmonic hypersurfaces in constant curvature spaces to be minimal or totally geodesic, which with parallel second fundamental form.