Higher-dimensional scalar–systolic inequality for real projective spaces

Determine whether every metric h on real projective space RP^m for m≥4 satisfies the scalar–systolic inequality (inf_{RP^m} Scal_h) sys(h)^2 ≤ m(m−1)π^2, with equality only for a round metric.

Background

The paper proves the scalar–systolic inequality for RP2 and RP3 and uses it to establish the sharp lower bound κ(F)2 ≥ 2m/(m+1) for smooth immersions of RPm into the unit Euclidean ball when m=2,3. The authors explicitly ask whether the same intrinsic inequality extends to dimensions m≥4 and whether its equality case is uniquely characterized by round metrics.

A positive resolution would extend the paper’s normal-curvature argument to all dimensions, yielding the sharp lower bound κ(F)2 ≥ 2m/(m+1) and the corresponding rigidity statement identifying the Veronese embedding as the equality case.

References

The above argument suggests the following question. Does the scalar--systolic inequality

\bigl(\inf_{RPm}Scal_h\bigr)sys(h)2 \leq m(m-1)\pi2

hold on $RPm$ for $m\geq4$, with equality only for the round metric?

— Normal Curvature and the Projective Systole  (2608.18002 - Chow et al., 18 Aug 2026) in Remark following the proof of Theorem 1