Veronese minimizes normal curvatures
Anton Petrunin
TL;DR
This work identifies an optimal normal-curvature bound for closed $n$-dimensional submanifolds $M$ in a large Euclidean ball: if the normal curvatures are strictly below $\tfrac{2}{\sqrt{3}}\cdot \tfrac{1}{r}$, then $M$ is homeomorphic to a sphere, while equality allows Veronese embeddings of projective planes $\RP^2$, $\CP^2$, $\HP^2$, or $\OP^2$ with all normal curvatures equal to $2$ realized on spheres of radius $r_n=\sqrt{n/(2n+2)}$. The approach combines a Bow lemma-based curvature comparison, a Gauss/Exponential-map analysis to decompose $M$ into discs, and a rigidity argument showing that the non-spherical border cases must arise from Veronese embeddings, yielding a sharp classification of extremal configurations. The results connect optimal curvature bounds to classical symmetric spaces and Veronese geometry, and raise open questions about minimality of the Veronese bound and the differentiable-type classification in borderline cases.
Abstract
Suppose M is a closed submanifold in a Euclidean ball of sufficiently large dimension. We give an optimal bound on the normal curvatures, guaranteeing that M is a sphere. The border cases consist of Veronese embeddings of the four projective planes.