On Embedded Spheres of Affine Manifolds

TL;DR

This paper proves that in closed affine manifolds with certain convex boundary conditions, the only embedded sphere that bounds a compact affine manifold is the standard ball, establishing a uniqueness result.

Contribution

It demonstrates that dome bodies in such affine manifolds are compact and that the standard ball uniquely bounds a compact affine manifold.

Findings

Dome bodies are shown to be compact.

A maximal dome body is a closed solid ball.

The standard ball bounds only one compact affine manifold.

Abstract

This paper studies certain embedded spheres in closed affine manifolds. For $n \geq 3$, we investigate the dome bodies in a closed affine $n$-manifold $M$ with its boundary homeomorphic to a sphere under the assumption that a developing map restricted to a component of $\partial\hat{M}$ is an embedding onto a strictly convex sphere in $\mathbb{A}^n$. By using the recurrent property of an incomplete geodesic we show that dome bodies are compact. Then a maximal dome body is a closed solid ball bounded by a component of $\partial\hat{M}$, and hence equals $\hat{M}$. The main theorem is that the standard ball in an affine space can only bound one compact affine manifold inside, namely the solid ball.