Combinatorial Gauss-Bonnet-Chern formula

A Stochastic Gauss-bonnet-chern Formula

We prove that a Gaussian ensemble of smooth random sections of a real vector bundle E over compact manifold M canonically defines a metric on E together with a connection compatible with it. Additionally, we prove a refined Gauss-Bonnet theorem stating that if the bundle E and the manifold M are oriented, then the Euler form of the above connection can be identified, as a current, with the expe...

A graph theoretical Gauss-Bonnet-Chern Theorem

We prove a discrete Gauss-Bonnet-Chern theorem ∑ g∈V K(g) = χ(G) for finite graphs G = (V,E), where V is the vertex set and E is the edge set of the graph. The dimension of the graph, the local curvature form K and the Euler characteristic are all defined graph theoretically.

The Gauss-Bonnet-Chern Theorem on Riemannian Manifolds

This expository paper contains a detailed introduction to some important works concerning the Gauss-Bonnet-Chern theorem. The study of this theorem has a long history dating back to Gauss’s Theorema Egregium (Latin: Remarkable Theorem) and culminated in Chern’s groundbreaking work [14] in 1944, which is a deep and wonderful application of Elie Cartan’s formalism. The idea and tools in [14] have...

A Gauss-Bonnet formula for closed semi-algebraic sets

We prove a Gauss-Bonnet formula for closed semi-algebraic sets.

The Gauss-Bonnet Theorem