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Introduction to Geometry and Topology
Inglese
109 b
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The book provides an introduction to topology, differential topology, and differential geometry. It is based on manuscripts that have been tested in different lecture cycles. The first chapter provides basic concepts and results from the theory of theorems. An exception to this is the Jordanian curvature, which is proved for polygons and a first idea of what kind of deeper topological problems are. In the second chapter, manifolds and groups are introduced and illustrated by a series of examples. Tangential and vector bundles, differentials, vector fields and Liesche brackets of vector fields are also discussed. This discussion is further deepened in the third chapter, in which de Rham's cohomology and the oriented integral are introduced, and the Brouwer's fixed point theorem, the Jordan-Brouwer's decomposition theorem, and the integral formula of Stokes. The final fourth chapter is devoted to the fundamentals of differential geometry. Along the development lines, which have traversed the geometry of the curves and submanifolds in Euclidean spaces, the connections and curvatures, the central concepts of differential geometry, are discussed. The Gaussian equations, the version of theorema egregium of Gauss for submanifolds of arbitrary dimension and codimension, form the climax. The book is mainly aimed at mathematics and physics students in the second and third year of studies and is suitable as a template for one- or two-semester lectures.
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