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

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 ${\mathrm{S}\mathrm{L}}_{\pm }(n+1,\mathbf{R})$. In this context, we present a concise proof demonstrating that ${\mathbf{R}\mathbf{P}}^n\mathrm{\#}{\mathbf{R}\mathbf{P}}^n$ and a few other manifolds do not possess a real projective structure when $n\ge 3$. Notably, our proof is shorter than those provided by Cooper-Goldman for $n=3$ and Çoban for $n\ge 4$. 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.