  Let $M^n, n\geq 3,$ be a complete Riemannian manifold of constant scalar
curvature $R$ and $f: M^n\rightarrow M^{n+k}(c)$ be an isometric immersion into
a space form with flat normal bundle. Assume that $f$ admits a principal normal
vector field of multiplicity $n-1$ at each point. Our main result is of a
global character and states that (i) $R\geq 0$ if $c= 0;$\ (ii) $ R>
(n-1)(n-2)c$ if $c> 0;$ and (iii) $R\geq n(n-1)c$ if $c< 0.$ The inequalities
are optimal. Our second result is of a local character and states that if in
addition we assume that $c$ is non-negative and the mean curvature field of $f$
is parallel then $M^n$ has non-negative sectional curvature.
