Curvatures and discrete Gauss-Codazzi equation in (2+1)-dimensional loop quantum gravity - Ariwahjoedi, Seramika; Kosasih, Jusak Sali; Rovelli, Carlo; Zen, Freddy (International journal of geometric methods in modern physics, 2015)
We derive the Gauss-Codazzi equation in the holonomy and plane-angle representations and we use the result to write a Gauss-Codazzi equation for a discrete (2+1)-dimensional manifold, triangulated by isosceles tetrahedra. This allows us to write operators acting on spin network states in (2+1)-dimensional loop quantum gravity, representing the 3-dimensional intrinsic, 2-dimensional intrinsic, and 2-dimensional extrinsic curvatures.