On the curvature of two-dimensional singular Regge lattices

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We consider geodesic approximation of a two-dimensional Riemannian manifold, M, with a singular Regge lattice K and, in particular, relations between classical continuum curvature power actions and corresponding lattice actions. By the singular Regge lattice we mean a triangulated piecewise flat space having very thin triangles. It is shown that the continuum actions are well approximated by the lattice actions in the sense of measures, provided that the edge lengths of K are small, independently of whether very thin triangles are contained in K or not.