arXiv subjects
Publications and source records attributed to Murat Limoncu.
In this note, using some regular triangular tilings of the sphere, the Euclidean plane and the hyperbolic plane, we examine the potential relationship between their discrete Bakry - Emery curvatures and the smooth curvatures of their ambient space forms.
We investigate whether there is a relationship between the discrete Bakry-Émery curvature of a graph and the smooth curvature of an ambient surface into which the graph is embedded geodesically. As we used weighted graphs as test objects, we developed a program for the calculation of the discrete curvature and with the help of this calculator, we observed some indications of such a relationship.
Comparison of $1$-dimensional distance functions is a basic tool in Alexandrov geometry and it is used to characterize spaces with curvature bounded above or below. For the zero curvature bound there is a differential inequality which enables one to check this comparison directly on a given smooth $1$-dimensional distance function. In this note we give a generalization of this property to arbitrary curvature bounds.