A sharp scalar curvature inequality for submanifolds (2402.05470v4)
Abstract: Let $Mn, n\geq 3,$ be a complete Riemannian manifold of constant scalar curvature $R$ and $f: Mn\rightarrow M{n+k}(c)$ 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 $Mn.$ 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 $Mn$ is bounded below by $c.$ As a consequence, we classify submanifolds which satisfy the latter condition.
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