If all geodesics are closed on the projective plane

TL;DR

This paper proves that the real projective plane has constant curvature if and only if all its geodesics are closed, establishing a unique geometric characterization related to G-structures.

Contribution

It demonstrates the equivalence between constant curvature and all geodesics being closed on RP2, and introduces the first known manifold with a single G-structure.

Findings

RP2 has constant curvature iff all geodesics are closed

If all geodesics are closed, then there are infinitely many simple closed geodesics

RP2 is the first known manifold with only one G-structure

Abstract

The paper shows that the curvature of RP2 is constant iff all geodesics are closed. Therefore RP2 is the first known manifold with only one G-structure. It took quiete a long time to find such a manifold. The author shows only that if all geodesics are closed then there are infinitely many simple closed geodesics. This proof is based on the geodesic return map and the theory of topological dynamics. From important results of Green, Grove and Gromoll one can conclude the theorem.