Complete immersions with constant scalar curvature and flat normal bundle

Let M be a complete Riemannian manifold with constant scalar curvature R and dimension n 3, and let f: M M^n+k (c) be a proper isometric immersion with flat normal bundle and type (1, n-1) in a complete simply-connected Riemannian space form M^n+k (c) of constant sectional curvature c and dimension n+k. Our first result is global and states that R 0 if c= 0, \ R> (n-1) (n-2) 2c if c> 0, and R n (n-1) 2c if c< 0. Our second result is of a local character and states that, if in addition we assume that c 0 and the mean curvature field of the immersion is parallel in the normal bundle, then M has non-negative sectional curvature.