Affine Projection Tensor Geometry: Decomposing the Curvature Tensor When the Connection is Arbitrary and the Projection is Tilted

TL;DR

This paper develops a general framework for projection tensor geometry on manifolds with arbitrary affine connections, removing previous restrictions and introducing new curvature tensors to analyze various physical and geometric structures.

Contribution

It introduces a comprehensive geometric approach for arbitrary projection tensors and affine connections, including new extrinsic and cross-projected curvature tensors, extending prior methods.

Findings

Defined two extrinsic curvature tensors that coincide for normal projections.

Introduced cross-projected curvature tensors for broader applications.

Applied the framework to general relativity and related physical theories.

Abstract

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.