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This paper constructs the geometrically natural objects which are associatedwith any projection tensor field on a manifold with any affine connection. Theapproaches to projection tensor fields which have been used in generalrelativity and related theories assume normal projection tensors ofco-dimension one and connections which are metric compatible and torsion-free.These assumptions fail for projections onto lightlike curves or surfaces andother situations where degenerate metrics occur as well as projections ontotwo-surfaces and projections onto spacetime in the higher dimensional manifoldsof unified field theories. This paper removes these restrictive assumptions.One key idea is to define two different ''extrinsic curvature tensors'' whichbecome equal for normal projections. In addition, a new family of geometricaltensors is introduced: the cross-projected curvature tensors. In terms of theseobjects, projection decompositions of covariant derivatives, the full Riemanncurvature tensor and the Bianchi identities are obtained and applied to perfectfluids, timelike curve congruences, string congruences, and the familiar 3+1analysis of the spacelike initial value problem of general relativity.Comment: 49 pages, RevTeX 3.

Affine projection tensor geometry: Decomposing the curvature tensor when the connection is arbitrary and the projection is tilted