Discrete curvature

Abstract

The combination of words ``discrete curvature'' is only an apparent contradiction. In this survey we describe curvature notions associated with polygons, polyhedral surfaces, and with abstract polyhedral manifolds. Several theorems about the discrete curvature are stated that repeat literally classical theorems of differential and Riemannian geometry: Theorema Egregium, Gauss--Bonnet theorem, and the Chern--Gauss--Bonnet theorem among the others. Some convergence results are also mentioned: under certain assumptions the discrete curvature tends to the smooth curvature as a smooth object is approximated by polyhedral ones.