Introduction to geometry and topology

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This book offers an introduction to topology, differential topology, and differential geometry, drawing from refined manuscripts used in various lecture courses. The first chapter presents elementary results and concepts from point-set topology, including a proof of the Jordan Curve Theorem for polygonal paths, which serves as an initial insight into deeper topological issues. The second chapter introduces manifolds and Lie groups, showcasing a variety of examples, and delves into tangent bundles, vector bundles, differentials, vector fields, and Lie brackets of vector fields. The third chapter expands on these topics by introducing de Rham cohomology and oriented integrals, providing proofs for the Brouwer Fixed-Point Theorem, the Jordan-Brouwer Separation Theorem, and Stokes's integral formula. The final chapter focuses on the fundamentals of differential geometry, tracing the evolution of concepts from curves to submanifolds of Euclidean spaces. Key topics include connections and curvature, culminating in the Gauß equations and a version of Gauß's theorema egregium applicable to submanifolds of any dimension and codimension. This book is designed for advanced undergraduates in mathematics and physics, serving as a foundation for a one- or two-semester bachelor's course.