Geometric moves relate geometric triangulations

TL;DR

This paper demonstrates that geometric triangulations of certain Riemannian manifolds are connected via geometric bistellar moves, with simplified relations in low dimensions, advancing understanding of triangulation transformations in differential geometry.

Contribution

It establishes the connectivity of geometric triangulations through bistellar moves for hyperbolic, spherical, and Euclidean manifolds, including simplified cases in dimensions 2 and 3.

Findings

Geometric triangulations are connected by bistellar moves after subdivisions.

In dimensions 2 and 3, triangulations are directly related without subdivisions.

The results apply to hyperbolic, spherical, and Euclidean manifolds.

Abstract

A geometric triangulation of a Riemannian manifold is a triangulation where the interior of each simplex is totally geodesic. Bistellar moves are local changes to the triangulation which are higher dimensional versions of the flip operation of triangulations in a plane. We show that geometric triangulations of a compact hyperbolic, spherical or Euclidean manifold are connected by geometric bistellar moves (possibly adding or removing vertices), after taking sufficiently many derived subdivisions. For dimensions 2 and 3, we show that geometric triangulations of such manifolds are directly related by geometric bistellar moves (without having to take derived subdivision).