On Embedded Spheres of Affine Manifolds

This paper studies certain embedded spheres in closed affine manifolds. For $n\ge 3$, 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 $\mathrm{\partial }\widehat{M}$ 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 $\mathrm{\partial }\widehat{M}$, and hence equals $\widehat{M}$. The main theorem is that the standard ball in an affine space can only bound one compact affine manifold inside, namely the solid ball.