Geometry of Manifolds and the Gauss-Bonnet Theorem

Abstract

This thesis delves into the fascinating realm of smooth manifolds, investigating their intri- cate geometry through a comprehensive exploration of tangent spaces, connections, curva- ture and the Gauss-Bonnet theorem, a crucial result in the theory of surfaces. This theorem establishes a connection between a surface’s geometric and topological properties and acts as a model for similar statements that hold true in higher-dimensional contexts. The study is organized into three chapters, each addressing key aspects of this subject and culminating with an illustrative example featuring the sphere.Chapter 1 serves as an introduction to the theory of differential manifolds. It begins with a comprehensive overview of the underlying concepts, including topological spaces, charts, atlases, and smooth maps. The chapter provides a detailed examination of tangent spaces and tangent bundles, shedding light on the notion of derivatives and directional derivatives on manifolds. Vector fields and vector bundles are then introduced, enabling a deeper understanding of the interplay between smooth functions and smooth vector fields. The chapter, after the introduction of tensor products, also explores the concept of a Rie- mannian metric, highlighting its significance in defining lengths, angles, and inner products on manifolds. Additionally, differentiable forms are discussed, unveiling their role in cap- turing the geometric properties of manifolds.