  This paper constructs the geometrically natural objects which are associated
with any projection tensor field on a manifold with any affine connection. The
approaches to projection tensor fields which have been used in general
relativity and related theories assume normal projection tensors of
co-dimension one and connections which are metric compatible and torsion-free.
These assumptions fail for projections onto lightlike curves or surfaces and
other situations where degenerate metrics occur as well as projections onto
two-surfaces and projections onto spacetime in the higher dimensional manifolds
of unified field theories. This paper removes these restrictive assumptions.
One key idea is to define two different ''extrinsic curvature tensors'' which
become equal for normal projections. In addition, a new family of geometrical
tensors is introduced: the cross-projected curvature tensors. In terms of these
objects, projection decompositions of covariant derivatives, the full Riemann
curvature tensor and the Bianchi identities are obtained and applied to perfect
fluids, timelike curve congruences, string congruences, and the familiar 3+1
analysis of the spacelike initial value problem of general relativity.
