Deforming convex projective manifolds

TL;DR

This paper investigates the deformation space of convex projective structures on manifolds with boundary, extending Koszul's theorem to non-compact cases and analyzing the openness of their holonomy representations.

Contribution

It generalizes Koszul's theorem to non-compact convex projective manifolds with boundary, establishing the openness of their holonomy representations in the representation variety.

Findings

Holonomies form an open subset in the representation variety for these manifolds.

Extension of Koszul's theorem to non-compact convex projective manifolds.

Analysis of convex ends and boundary conditions in the deformation theory.

Abstract

We study a properly convex real projective manifold with (possibly empty) compact, strictly convex boundary, and which consists of a compact part plus finitely many convex ends. We extend a theorem of Koszul which asserts that for a compact manifold without boundary the holonomies of properly convex structures form an open subset of the representation variety. We also give a relative version for non-compact (G,X)-manifolds of the openess of their holonomies.