### Abstract:

A standard tool in general relativity is the 3+1 or ADM point of view, namely
slicing spacetime into spacelike hypersurfaces of constant time and then describing
physics in terms of time-dependent quantities on a typical such hypersurface. Much
less well-known is the 1+3 point of view, in which one foliates spacetime with
timelike curves, then describes physics in terms of the surfaces "locally orthogonal"
to the given foliation. This is precisely the description of physics as seen by a
single observer. However, in many instances there do not exist such orthogonal
hypersurfaces. One may instead attempt to describe physics on the manifold of
orbits defined by the timelike curves, but one must then develop a parametric
theory to handle the time dependent objects defined on the manifold of orbits.
I will present two equivalent descriptions of parametric manifolds. The first
is based on a generalized Gauss-Codazzi formalism which involves projection to a
lower-dimensional "surface". The second is an intrinsic description which involves
redefining the action of vector fields on functions. In either description one is lead
to generalized notions of connections, Lie bracket, and exterior differentiation.
Unique to a parametric theory of geometry is the deficiency. Although independent
of the torsion, the deficiency behaves like torsion in the parametric direction.
We will show how the deficiency emerges as a result of the above generalizations. The 3+1 formalism arises naturally in considering initial-value formulations
both for fields on a fixed background spacetime and for the spacetime itself. The
applicability of parametric manifolds to such problems will be discussed.