$\mathbf{RP}^n \# \mathbf{RP}^n$ and some others admit no real projective structure

Abstract

A manifold $M$ possesses a real projective structure if it has an atlas consisting of charts mapping to $\mathbf{S}^n$, where the transition maps lie in $\mathrm{SL}_\pm(n+1, \mathbf{R})$. In this context, we present a concise proof demonstrating that $\mathbf{RP}^n\#\mathbf{RP}^n$ and a few other manifolds do not possess a real projective structure when $n\geq3$. Notably, our proof is shorter than those provided by Cooper-Goldman for $n=3$ and \c{C}oban for $n\geq 4$. To do this, we reprove the classification of closed real projective manifolds with infinite-cyclic holonomy groups by Benoist due to a small error. We will leverage the concept of the octantizability of real projective manifolds with nilpotent holonomy groups, as introduced by Benoist and Smillie, which serves as a powerful tool.