Petrunin’s question on real projective spaces

Let $m\ge 2$ , let $g$ be a Riemannian metric on $\mathbb{R}\mathbb{P}^{m}$ , and let $\operatorname{sys}(g)$ denote the length of the shortest loop representing the nontrivial element of $\pi_{1}(\mathbb{R}\mathbb{P}^{m})$ . If $g_{\mathrm{can}}$ is the standard metric of constant sectional curvature $1$ , is it true that

\operatorname{Vol}_{g}(\mathbb{R}\mathbb{P}^{m})\ge \operatorname{Vol}_{g_{\mathrm{can}}}(\mathbb{R}\mathbb{P}^{m})\left(\frac{\operatorname{sys}(g)}{\pi}\right)^{m}?

Moreover, does equality hold only when $g$ has constant sectional curvature, equivalently when $g$ is homothetic to the standard metric?

Progress summary

Solved

A new preprint reports that the first unresolved three-dimensional case is solved, extending the known two-dimensional result and making the equality case rigid.

Petrunin’s question asks for the sharp projective-systolic inequality and characterization of equality on real projective spaces. The scan records results for dimensions $2$ and $3$ only.

Known results

On $\mathbb{RP}^2$ , the canonical metric is determined up to isometry by its volume spectrum, using rigidity, Weinstein’s theorem, and equality in Pu’s inequality.

August 2026 confirmation in $\mathbb{RP}^3$

A new arXiv preprint reports the first open case as proved: sharp systolic inequalities establish the $\mathbb{RP}^3$ result and rigidify equality, forcing constant sectional curvature. This is a preprint-level confirmation, with no separate referee report or independent exposition found in the scan.

Current status (as of August 2026): The cases $\mathbb{RP}^2$ and $\mathbb{RP}^3$ are reported as settled, while no conclusion is recorded for higher dimensions.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

arXiv

Additional references

Normal Curvature and the Projective Systole — arXiv

differential-geometry geometric-analysis

2 days ago