A sharp scalar curvature inequality for submanifolds
H. A. Gururaja
TL;DR
This work addresses curvature constraints for complete submanifolds $M^n$ of space forms under a flat normal bundle and a principal normal of multiplicity $n-1$. It derives a global scalar-curvature inequality linking $R$ and the ambient curvature $c$, with sharp bounds in the three regimes $c=0$, $c>0$, and $c<0$, and proves these results globally under completeness. Additionally, a parallel mean-curvature condition yields a sectional-curvature lower bound $K\ge c$, enabling a local result and a classification of submanifolds in the nonnegative ambient curvature case. The approach hinges on adapted principal coordinates, convexity along geodesics, and the key quantities $D=\frac{R}{2(n-1)}$ and $\phi=c+\langle \eta_1, \eta_2\rangle$, extending known hypersurface bounds to higher codimension with flat normal bundles.
Abstract
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 which has multiplicity $n-1$ at each point of $M^n.$ Our first result is global 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 $c< 0.$ These inequalities are optimal. Our second result states that if we further assume that the mean curvature field of $f$ is parallel, then the sectional curvature of $M^n$ is bounded below by $c.$ As a consequence, we classify submanifolds which satisfy the latter condition.