Gauss--Bonnet theorem for compact and orientable surfaces: a proof without using triangulations

TL;DR

This paper offers an intrinsic proof of the Gauss--Bonnet theorem for compact, orientable surfaces using complex structures, avoiding the traditional triangulation approach.

Contribution

It introduces a novel proof method for the Gauss--Bonnet theorem that relies solely on complex structures, bypassing triangulation techniques.

Findings

Proof is purely intrinsic and avoids triangulations

Utilizes complex structures to establish the theorem

Provides a new perspective on classical differential geometry results

Abstract

The aim of this note is to provide an intrinsic proof of the Gauss--Bonnet theorem without invoking triangulations, which is achieved by exploiting complex structures.