  A parametric manifold can be viewed as the manifold of orbits of a (regular)
foliation of a manifold by means of a family of curves. If the foliation is
hypersurface orthogonal, the parametric manifold is equivalent to the
1-parameter family of hypersurfaces orthogonal to the curves, each of which
inherits a metric and connection from the original manifold via orthogonal
projections; this is the well-known Gauss-Codazzi formalism. We generalize this
formalism to the case where the foliation is not hypersurface orthogonal.
Crucial to this generalization is the notion of deficiency, which measures the
failure of the orthogonal tangent spaces to be surface-forming, and which
behaves very much like torsion. Some applications to initial value problems in
general relativity will be briefly discussed.
