In 2002, LeBrun and Mason showed that the only projective structure on real projective two-space with all geodesics being simple closed curves is the standard one. I shall show that the corresponding infinitesimal statement holds in all dimensions. This boils down to identifying the kernel of a suitable X-ray transform. I’ll explain how this goes, starting from scratch. This is real integral geometry on real projective space: Funk transform, Radon transform, and all that. I’ll also explain how one might approach similar questions on complex projective space.