Normal Curvature and the Projective Systole · arXivDeskAbstract
For a smooth immersion F:RPm↬BN(1), we observe that the sharp systolic inequality forces a sharp lower bound on its maximal normal curvature κ(F)
. In dimensions
, the sharp inequalities of Pu and Bray--Brendle--Eichmair--Neves therefore give
κ(F)2≥m+12m . Equality holds precisely for the Veronese embedding. This recovers Petrunin's theorem for
RP2 and, for
RP3 , confirms the first open case of his question for real projective spaces.