Closed and complete spacelike hypersurfaces in Minkowski space

Let M be an embedded submanifold of a complete Riemannian manifold V. If M is a closed subset, then it must also be complete. However, the converse is false, even if V is Rsup(n). The situation is quite different for spacelike hypersurfaces in Lorentz manifolds. It is shown that for f:Msup(n) → Lsup(n + 1), a spacelike immersion into Minkowski space, if f induces a complete metric on M then f(M) is closed; indeed, f is an embedding, and f(M) is an achronal graph over any spacelike hyperplane. Conversely, it is shown that for Vsup(n + 1) a b-complete Lorentz manifold and f:Msup(n) → Vsup(n + 1), a spacelike immersion with bounded principal curvatures, if f is proper then it induces a complete metric on M. (author)