Normal Curvature and the Projective Systole

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