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→ Mⁿ⁺ᵏ(c) be a proper isometric immersion with flat normal bundle and type $(1, n-1)$ in a complete simply-connected Riemannian space form Mⁿ⁺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.