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. Chapter 2 focuses on the study of covariant derivatives, connections, and curvature on Riemannian manifolds. Covariant derivatives generalize the notion of differentiation to curved spaces, allowing for the introduction of parallel transport. The chapter delves into the properties and construction of connections, exploring their local and global aspects, highlighting the role they play in defining parallel transport, and revealing their crucial role in measuring differentiation along curves on manifolds. In particular, we focus on the ”Levi-Civita connection”, which is the unique connection on Riemannian manifolds that is metric-compatible and is symmetric. Moreover, the curvature associated with these con- nections is investigated, unveiling the geometric information encoded within the Riemann curvature tensor. The concepts of sectional curvature is explored, emphasizing its signif- icance in characterizing the intrinsic geometry of Riemannian manifolds. And finally, we introduce the Gauss-Bonnet theorem, a classical result which highlights the interaction be- tween the topology and geometry of surfaces. In Chapter 3, the theoretical framework established in the previous chapters is applied to a specific example - the 2-sphere. By considering the sphere as a smooth Riemannian manifold, the chapter presents a detailed analysis of its geometry, connections, and curva- ture. The intrinsic properties of the sphere, such as the its (round) metric, the Levi-Civita connection, and the associated curvature tensors, are examined in depth. In the final section of this chapter, we apply the Gauss-Bonnet formula to compute the Euler characteristic of the 2-sphere, a topological invariant using the knowledge of the curvature which is a geo- metric data. The content of this last chapter is due to the independent work of the author of this thesis. ix Through the systematic exploration of tangent spaces, connections, and curvature on smooth manifolds, this thesis deepens our understanding of their geometric properties. By providing a concrete example through the study of the 2-sphere, it not only serves as a testament to the broader field of differential geometry but also paves the way for further investigations into the rich world of smooth manifolds