  We use a construction which we call generalized cylinders to give a new proof
of the fundamental theorem of hypersurface theory. It has the advantage of
being very simple and the result directly extends to semi-Riemannian manifolds
and to embeddings into spaces of constant curvature. We also give a new way to
identify spinors for different metrics and to derive the variation formula for
the Dirac operator. Moreover, we show that generalized Killing spinors for
Codazzi tensors are restrictions of parallel spinors. Finally, we study the
space of Lorentzian metrics and give a criterion when two Lorentzian metrics on
a manifold can be joined in a natural manner by a 1-parameter family of such
metrics.
