Some curvature inequalities for submanifolds

In this paper, we first study isometric immersions $f:M^n\to M^{n+k}(c),n\ge 3,$ into space forms with flat normal bundle and constant scalar curvature $R\mathrm{.}$ Under a suitable multiplicity condition on the second fundamental form of $f,$ we prove the global result that $R>(n-1)(n-2)c$ if $c>0,$ and $R\ge n(n-1)c$ if $c\le 0.$ We then obtain a classification for submanifolds with flat normal bundle having constant scalar and mean curvatures. Finally, in the hypersurface case, analogous results are proved for conformally flat hypersurfaces with constant higher order mean curvatures.