(\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.