The smoothness of the real projective deformation spaces of orderable Coxeter 3-polytopes

TL;DR

This paper proves that the deformation space of certain orderable Coxeter 3-polytopes in real projective space forms a smooth manifold, providing insights into their geometric structure and deformation properties.

Contribution

It establishes the smoothness of the deformation space for orderable Coxeter 3-polytopes of normal type, a significant advancement in understanding their geometric moduli.

Findings

Deformation space is a smooth manifold under specified conditions.

Orderability and normal type are key for smoothness.

Provides a framework for analyzing Coxeter polytope deformations.

Abstract

A Coxeter polytope is a convex polytope in a real projective space equipped with linear reflections in its facets, such that the orbits of the polytope under the action of the group generated by the linear reflections tessellate a convex domain in the real projective space. Vinberg proved that the group generated by these reflections acts properly discontinuously on the interior of the convex domain, thus inducing a natural orbifold structure on the polytope. In this paper, we consider labeled combinatorial polytopes $\mathcal{G}$ associated to such orbifolds, and study the deformation space $\mathcal{C} (\mathcal{G})$ of Coxeter polytopes realizing $\mathcal{G}$. We prove that if $\mathcal{G}$ is orderable and of normal type then the deformation space $\mathcal{C}(\mathcal{G})$ of real projective Coxeter 3-polytopes realizing $\mathcal{G}$ is a smooth manifold. This result is achieved by analyzing a natural map of $\mathcal{C} (\mathcal{G})$ into a smooth manifold called the realization space.