Pseudo-Riemannian geometry on a simplicial lattice and the extrinsic curvature tensor

We define the simplicial analogues of two concepts from differential topology: the concept of a point on the simplicial manifold and the concept of a tangent space on a simplicial manifold. We derive the simplicial analogues of parallel transport, the covariant derivative, connections, the Riemann curvature tensor, and the Einstein tensor. We construct the extrinsic curvature for a simplicial hypersurface using the simplicial covariant derivative. We discuss the importance of this simplicial extrinsic curvature to the 3+1 Regge-calculus program. It appears to us that the newly developed null-strut lattice is the most natural version of a 3+1 Regge lattice for the construction of extrinsic curvature. (A null-strut lattice is a 3+1 Regge spacetime lattice with TrK=const simplicial hypersurfaces, each connected to its two adjacent hypersurfaces entirely by simplicial light cones built of null struts.) Finally, we test the Regge-calculus version of the extrinsic curvature on a Bianchi type-IX simplicial hypersurface. The calculation agrees with the continuum expression to first order.