The Space of Strictly-convex Real-projective structures on a closed manifold

TL;DR

This paper provides an expository proof that the space of holonomies of strictly-convex real-projective structures on a closed manifold forms a subset of the representation space that is both open and closed, highlighting its topological properties.

Contribution

It offers a clear proof that the set of holonomies for strictly-convex real-projective structures on a closed manifold is topologically well-behaved within the representation space.

Findings

The set of holonomies is both open and closed in the representation space.

Strictly-convex real-projective structures have a topologically stable holonomy set.

The proof clarifies the structure of the deformation space of such geometric structures.

Abstract

This is an expository proof that, if $M$ is a compact $n$-manifold with no boundary, then the set of holonomies of strictly-convex real-projective structures on $M$ is a subset of $\operatorname{Hom}(\pi_1M,\operatorname{PGL}(n+1,\mathbb R))$ that is both open and closed.