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

A manifold $M$ possesses an $\mathbf{RP}^n$-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 an $\mathbf{RP}^n$-structure when $n\geq3$. Notably, our proof is shorter than those provided by Cooper-Goldman for $n=3$ and Coban for $n\geq 4$. To do this, we reprove the classification of closed $\mathbf{RP}^n$-manifolds with infinite cyclic holonomy groups by Benoist due to a small error. We will leverage the concept of octantizability of $\mathbf{RP}^n$-manifolds with nilpotent holonomy groups, as introduced by Benoist and Smillie, which serves as a powerful tool.
Comment: 16 pages. We shortened the paper and revised some technical errors