Geodesic completeness of spacelike and timelike submanifolds of Minkowski space \(L^ n\) are considered in this paper. A key property used is that the Levi-Civita connection of Minkowski space is the same as the standard connection for \({\mathbb{R}}^ n\) [cf. S. G. Harris and K. Nomizu, ibid. 13, 347-350 (1983; Zbl 0512.53027)].
The first class of results of the present paper concerns properly immersed spacelike submanifolds of arbitrary codimension in \(L^ n\). If a suitably chosen normal field to M satisfies a subaffine growth condition, then the given submanifold is geodesically complete. Analogous result is obtained for geodesics in a timelike hypersurface for which additionally the second fundamental form operator S(c’,c’) is bounded. Finally for proper timelike hypersurfaces with diagonalizable second fundamental forms, conditions are given on the principal curvatures which guarantee past and future completeness of a given nonspacelike geodesic.