  Submanifolds in Lorentz-Minkowski space are investigated from various
mathematical viewpoints and are of interest also in relativity theory. We
define the hyperbolic surface and the de Sitter surface of a curve in the
spacelike hypersurface M in the Minkowski 4-space. These surfaces are
respectively located in the hyperbolic 3-space and in the de Sitter 3-space. We
use techniques of the theory of singularities in order to describe the generic
shape of these surfaces and of their singular value sets. We also investigate
geometric meanings of those singularities. The results in this paper contribute
to the study of the extrinsic geometry of curves in different ambient spaces.
