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 $\kappa(F)^2\ge 2m/(m+1)$, with equality only for the Veronese embedding, up to congruence. We also prove that a closed, connected, orientable three-manifold admitting an immersion into a Euclidean unit ball with $\kappa(F)\le\sqrt{3/2}$ is diffeomorphic to $S^3$, $\mathbb{R}\mathbb{P}^3$, or $S^2\times S^1$. All three possibilities occur, while $\kappa(F)<\sqrt{3/2}$ forces $X\cong S^3$. These results answer a question of Petrunin and prove a conjecture of Chodosh--Li concerning the normal curvature of three-manifolds. The key intrinsic input is the strict scalar--systolic inequality \[ (\min_Y R_g)\text{sys}(g)^2<6\pi^2 \] for every spherical three-space form $Y$ with $|\pi_1(Y)|>2$. Its proof uses systolic monotonicity along Ricci flow with surgery. This strict inequality complements the sharp scalar--systolic inequality for $\mathbb{R}\mathbb{P}^3$ of Bray--Brendle--Eichmair--Neves.