  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 will classify closed $\mathbf{RP}^n$-manifolds
with infinite cyclic holonomy groups based on Benoist's work. We will leverage
the concept of $n$-octantizability of $\mathbf{RP}^n$-manifolds with nilpotent
holonomy groups, as introduced by Benoist and Smillie, which serves as a
powerful tool.
