Isometric or harmonic mappings of complete Riemannian manifolds

We investigate (a) the isometric and (b) the harmonic mappings phi of a complete Riemannian manifold M-n whose sectional curvature is bounded from below, into euclidean space En+m and in case of (a) also into the unit sphere Sn+m-1 subset of En+m. In case of (a) we obtain conditions in terms of the euclidean norm parallel toH(phi(x))parallel to x epsilon M-n of the mean curvature vector of phi(M-n) on the radius r of the euclidean ball B(r) in order that phi(M-n) cannot be pinched in any such B(r) (phi(M-n) not subset of B(r)). In case of (b) we show that under a mild condition on the Ricci curvature the positivity of the energy density e(phi) is necessary in order that phi(M-n) spreads out to infinity.