Convex real projective structures and Weil's local rigidity Theorem

TL;DR

This paper investigates the deformation space of convex real projective structures on hyperbolic manifolds, calculating tangent spaces and proving local rigidity results for hyperbolic lattices.

Contribution

It computes the Zariski tangent space at Fuchsian loci in character varieties and establishes Weil's local rigidity theorem for hyperbolic lattices using real projective structures.

Findings

Tangent space at Fuchsian loci consists of cubic forms.

Proved Weil's local rigidity theorem for hyperbolic lattices.

Connected real projective structures with hyperbolic geometry.

Abstract

For an $n$-dimensional real hyperbolic manifold $M$, we calculate the Zariski tangent space of a character variety $\chi(\pi_1(M),SL(n+1,\mathbb R)), n>2$ at Fuchisan loci to show that the tangent space consists of cubic forms. Furthermore we prove the Weil's local rigidity theorem for uniforml hyperbolic lattices using real projective structures.