The manifold of polygons degenerated to segments

Abstract

In this paper we study the space $L\left(n\right)$ of n-gons in the plane degenerated to segments. We prove that this space is a smooth real submanifold of $C^n$, and describe its topology in terms of the manifold $M\left(n\right)$ of n-gons degenerated to segments and with the first vertex at 0. We show that $M\left(n\right)$ and $L\left(n\right)$ 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 $L\left(n\right)$ by the diagonal action of the affine complex group and the re-enumeration of the vertices is described.