Normal Curvature and the Projective Systole

For a smooth immersion $F:\mathbb{R}{\mathbb{P}}^m\looparrowright {\bar{\mathbb{B}}}^N(1)$, we observe that the sharp systolic inequality forces a sharp lower bound on its maximal normal curvature $\kappa (F)$. In dimensions $m=2,3$, the sharp inequalities of Pu and Bray–Brendle–Eichmair–Neves therefore give $\kappa (F{)}^2\ge \frac{2m}{m+1}$. Equality holds precisely for the Veronese embedding. This recovers Petrunin’s theorem for $\mathbb{R}{\mathbb{P}}^2$ and, for $\mathbb{R}{\mathbb{P}}^3$, confirms the first open case of his question for real projective spaces.