On complete submanifolds with parallel mean curvature in negative pinched manifolds

A rigidity theorem for oriented complete submanifolds with parallel mean curvature in a complete and simply connected Riemannian (n + p)- dimensional manifold Nn+p with negative sectional curvature is proved. For given positive integers n (≥ 2), p and for a constant H satisfying H > 1 there exists a negative number τ (n, p, H) ε (-1,0) with the property that if the sectional curvature of N is pinched in [-1, ε(n, p, H)], and if the squared length of the second fundamental form is in a certain interval, then Nn+p is isometric to the hyperbolic space Hn+p(-1). As a consequence, this submanifold M is congruent to Sn (1/√H2 - 1) or the Veronese surface in S4 (1/√H2 - 1). © Editorial Committee of Applied Mathematics 2007.