We determine maximal space-like hypersurfaces which are the images of subbundles of the normal bundle of m-dimensional totally geodesic space-like submanifolds of an $(m+2)$-dimensional Lorentzian space form ${widetilde M}^{m+2}_{1}(c)$ under the normal exponential map. Then we construct examples of maximal space-like hypersurfaces of ${widetilde M}^{m+2}_{1}(c)$}.

Keywords: Maximal hypersurface, space-like submanifold, mean curvature, Lorentzian space form