We introduce new definitions of sectional, Ricci, and scalar curvatures for networks and their higher dimensional counterparts, derived from two classical notions of curvature for curves in general metric spaces, namely, the Menger curvature and the Haantjes curvature. These curvatures are applicable to unweighted or weighted and undirected or directed networks and are more intuitive and easier to compute than other network curvatures. In particular, the proposed curvatures based on the interpretation of Haantjes definition as geodesic curvature allow us to give a network analogue of the classical local Gauss–Bonnet theorem. Furthermore, we propose even simpler and more intuitive proxies for the Haantjes curvature that allow for even faster and easier computations in large-scale networks. In addition, we also investigate the embedding properties of the proposed Ricci curvatures. Lastly, we also investigate the behavior, both on model and real-world networks, of the curvatures introduced herein with more established notions of Ricci curvature and other widely used network measures.

We introduce a metric notion of Ricci curvature forPLmanifolds and study its convergence properties. We also prove a fitting version of the Bonnet-Myers theorem, for surfaces as well as for a large class of higher dimensional manifolds.

A new Combinatorial Ricci curvature and Laplacian operators for grayscale images are introduced and tested on 2D synthetic, natural and medical images. The effectiveness of the notions introduced herein, as compared with more classical methods, is shown by a variety of experimental results. Analogue formulae for voxels are also obtained. This novel approach is based upon more general concepts developed by R. Forman. Further applications, in particular a fitting Ricci flow, are discussed.