Vector Bundles and Connections

This textbook offers a self-contained introduction to the theory of connections on vector bundles that is accessible to both advanced undergraduate students and graduate students. Constructions and proofs of key results are presented in detail in order to be easily understandable and instructive, and each chapter concludes with a set of interesting exercises. Standard material about vector bundles is covered in the first chapter, with many examples illustrating the main concepts. Chapter 2 is concerned with the theory of connections on vector bundles, with special attention to the curvature of a connection. The third chapter explores several useful topics not always included in similar texts, such as the computation of the holomorphic tangent and canonical bundles of a Grassmann manifold and the curvature of the tautological and tautological quotient bundles. Finally, Chapter 4 discusses Chern, Pontryagin and Euler classes as an important application of the theory of connections on vector bundles to the theory of characteristic classes. This book can serve as a text for a one-semester course in differential geometry focused on vector bundles and connections, or as a resource for students pursuing studies in algebraic geometry and mathematical physics. Readers should have a basic understanding of manifolds, differential forms, and cohomology.