On embedded spheres of affine manifolds

Abstract

This paper studies certain embedded spheres in closed affine manifolds. For n greater than or equal to 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 the boundary of hat{M} is an embedding onto a strictly convex sphere in 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 the boundary of 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.