On the two-systole of real projective spaces

TL;DR

This paper characterizes the largest two-systole in real projective spaces within conformal classes using an integral-geometric approach to minimal two-spheres in homogeneous three-spheres.

Contribution

It introduces a new integral-geometric formula and uniquely identifies the metric maximizing the two-systole in each conformal class of real projective spaces.

Findings

Identifies the metric with the largest two-systole in each conformal class.

Provides an integral-geometric formula for minimal two-spheres.

Characterizes homogeneous metrics on real projective spaces.

Abstract

We establish an integral-geometric formula for minimal two-spheres inside homogeneous three-spheres, and use it to provide a characterisation of each homogeneous metric on the three-dimensional real projective space as the unique metric with the largest possible two-systole among metrics with the same volume in its conformal class.