Abstract
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 Rn. The situation is quite different for spacelike hypersurfaces in Lorentz manifolds. It is shown that for f:Mn to Ln+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 Vn+1 a b-complete Lorentz manifold and f:Mn to Vn+1, a spacelike immersion with bounded principal curvatures, if f is proper (e.g. a closed embedding) then it induces a complete metric on M.
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