Geodesic completeness of submanifolds in Minkowski space

A nondegenerate immersed submanifold M of Minkowski space L" has an induced metric which is either positive definite or Lorentzian. In the former case M is said to be spacelike and in the latter M is timelike. Spacelike hypersurfaces have played an important role in General Relativity in numerical calculations and in singularity theorems. For example, several singularity theorems of Galloway and Frankel [5] require that the given space-time contain a complete spacelike hypersurface. By the Hopf-Rinow theorem, a closed imbedded submanifold of a complete Riemannian manifold is automatically geodesically complete. But the situation is more complicated in the Lorentzian setting. Properly imbedded (hence closed) nondegenerate submanifolds of L" may fail to be geodesically complete (cf. Harris [9]). This holds for spacelike and timelike submanifolds of any codimension from 1 to n- 1. Thus, further hypotheses than 'closed' or 'proper' must be imposed on nondegenerate Lorentzian submanifolds to guarantee geodesic completeness of the given submanifold. Cheng and Yau [2] have shown that a closed imbedded spacelike hypersurface in L" is complete if it has constant mean curvature. Harris [9] has established the completeness of properly immersed spacelike hypersurfaces with bounded principal curvatures in b-complete space-times. In the first part of the present paper we prove that a properly immersed spacelike submanifold in L" is complete if a minimal unit timelike normal field satisfies a subaffine growth condition. This is a sufficient condition for properly immersed spacelike submanifolds of arbitrary codimension to be geodesically complete. As for as we know, it is the first result on completeness of spacelike submanifolds of arbitrary codimension. Extensive studies of isometric immersions of indefinite space forms as hypersurfaces of higher dimensional indefinite space forms have recently been made (cf. [3], [6], [7], [8], and [12]). It is then natural to seek sufficient conditions for geodesic completeness of timelike submanifolds. The study of timelike submanifolds is somewhat more complicated than spacelike submanifolds since the induced metric is not definite. In particular, the second fundamental form of a timelike hypersurface is unbounded on unit vectors