EULER CHARACTERISTIC AND SURFACES

The Euler characteristic is a topological notion appearing in many different topics throughout mathematics. The Euler formula shows that the expression of the Euler characteristic in terms of numbers of vertices, edges and regions (faces of different dimension in higher- dimensional cases) is a topological invariant. I had firstly come across the topic by reading the topological invariants. This formula is a powerful tool used in establishing many important mathematical results, from the classification of regular polyhedra to the nonplanarity criterion for graphs. I will describe some required tools in differential geometry. Gauss- Bonnet is also a deep result in differential geometry that illustrates a fundamental relationship between the curvature of a surface and its Euler characteristic. In this paper I introduce and examine properties of surfaces in effort to prove a discrete Gauss Bonnet analog.