The manifold of polygons degenerated to segments

TL;DR

This paper investigates the geometric structure of the space of polygons degenerated to segments, establishing it as a smooth manifold, analyzing its topology, geodesics, and symmetries.

Contribution

It characterizes the space of degenerated polygons as a smooth manifold, describes its topology, and computes geodesic equations and symmetry quotients.

Findings

The space of degenerated polygons forms a smooth real submanifold of ^n.

Explicit geodesic equations are derived for these manifolds.

The quotient space under affine and re-enumeration actions is described.

Abstract

In this paper we study the space $\mathbb{L}(n)$ of $n$-gons in the plane degenerated to segments. We prove that this space is a smooth real submanifold of $\mathbb{C}^n$, and describe its topology in terms of the manifold $\mathbb{M}(n)$ of $n$-gons degenerated to segments and with the first vertex at 0. We show that $\mathbb{M}(n)$ and $\mathbb{L}(n)$ contain straight lines that form a basis of directions in each one of their tangent spaces, and we compute the geodesic equations in these manifolds. Finally, the quotient of $\mathbb{L}(n)$ by the diagonal action of the affine complex group and the re-enumeration of the vertices is described.