Small Normal Curvature and Three-Manifold Topology

For $m=2,3$, we prove that every smooth immersion $F:\mathbb{R}\mathbb{P}^m\looparrowright\overline{\mathbb B}^{,N}(1)$ satisfies $κ(F)^2\ge 2m/(m+1)$, with equality only for the Veronese embedding, up to congruence. We also prove that a closed, connected, orientable three-manifold admitting an imme