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

TL;DR

The paper provides a concise proof that certain high-dimensional manifolds, including the connected sum of two real projective spaces, do not admit a real projective structure, improving on previous proofs in length and methodology.

Contribution

It offers a shorter, more streamlined proof of non-existence of real projective structures on specific manifolds, revisiting and correcting classification results for manifolds with infinite-cyclic holonomy.

Findings

$ extbf{RP}^n extbf{ extlangle} extbf{RP}^n$ and others admit no real projective structure for $n \\geq 3$

Shorter proof compared to previous works by Cooper-Goldman and Coban

Revises classification of manifolds with infinite-cyclic holonomy groups using Benoist's work

Abstract

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 $\mathrm{SL}_\pm(n+1, \mathbf{R})$. In this context, we present a concise proof demonstrating that $\mathbf{RP}^n\#\mathbf{RP}^n$ and a few other manifolds do not possess a real projective structure when $n\geq3$. Notably, our proof is shorter than those provided by Cooper-Goldman for $n=3$ and \c{C}oban for $n\geq 4$. 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.