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A manifold $M$ possesses an $\mathbf{RP}^n$-structure if it has an atlasconsisting of charts mapping to $\mathbf{S}^n$, where the transition maps liein $\mathrm{SL}_\pm(n+1, \mathbf{R})$. In this context, we present a conciseproof demonstrating that $\mathbf{RP}^n\#\mathbf{RP}^n$ and a few othermanifolds 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 Cobanfor $n\geq 4$. To do this, we will classify closed $\mathbf{RP}^n$-manifoldswith infinite cyclic holonomy groups based on Benoist's work. We will leveragethe concept of $n$-octantizability of $\mathbf{RP}^n$-manifolds with nilpotentholonomy groups, as introduced by Benoist and Smillie, which serves as apowerful tool.

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