Combinatorial curvature is defined for polyhedral graphs analogue to Gaussian curvature for Riemannian 2-manifolds. Some aspects of polyhedral graphs from the perspective of combinatorial curvature will be studied in this thesis.

Basics in Gaussian curvature, Euler characteristic and gluing as a method to construct 2-manifolds are revised in Chapter 1. Definition and properties of combinatorial curvature are revised in Chapter 2. Theorems by DeVos and Mohar, Chen and Chen, Baues and Peyerimhoff as well as Stone analogous to those in Riemannian geometry are reviewed. In particular, the inadequacy in the proof of Stone is analyzed.

In Chapter 3, polygonal surfaces are constructed from polyhedral graph. This can be viewed as kind of deformation through re-distribution of curvature. The p...[ Read more ]

Combinatorial curvature is defined for polyhedral graphs analogue to Gaussian curvature for Riemannian 2-manifolds. Some aspects of polyhedral graphs from the perspective of combinatorial curvature will be studied in this thesis.

Basics in Gaussian curvature, Euler characteristic and gluing as a method to construct 2-manifolds are revised in Chapter 1. Definition and properties of combinatorial curvature are revised in Chapter 2. Theorems by DeVos and Mohar, Chen and Chen, Baues and Peyerimhoff as well as Stone analogous to those in Riemannian geometry are reviewed. In particular, the inadequacy in the proof of Stone is analyzed.

In Chapter 3, polygonal surfaces are constructed from polyhedral graph. This can be viewed as kind of deformation through re-distribution of curvature. The polygonal surfaces are proved to be Alexandrov spaces. Alexandrov space, in which singularities are allowed, is a generalization of Riemannian manifold. Lipschitz relations among metrics in the different spaces as well as curvature estimates are established. The big face obstacle for polyhedral graphs is also eliminated. Consequently combinatorial Bonnet-Myers Theorems are proved.

During the constructions, a kind of curvature flow is discovered and proved useful. In addition, this curvature flow shows a direction of generalization of the combinatorial Bonnet-Myers Theorems.

In Chapter 4, interesting questions for furher investigations are discussed.