Transcription of Introduction to Differential Geometry
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Introduction to Differential Geometry
Introduction to Differential GeometryLecture Notes for MAT367 Contents1 Introduction .. Some history .. The concept of manifolds : Informal discussion .. manifolds in Euclidean space .. Intrinsic descriptions of manifolds .. Surfaces ..62 manifolds .. Atlases and charts .. Definition of manifold .. Examples of manifolds .. projective spaces .. projective spaces .. Grassmannians .. Oriented manifolds .. Open subsets .. Compact subsets .. Appendix .. relations .. 333 Smooth maps.. Smooth functions on manifolds .. Smooth maps between manifolds .. of manifolds .. Examples of smooth maps .. , diagonal maps .. diffeomorphismRP1 =S1.. diffeomorphismCP1 =S2.. to and from projective space .. quotient mapS2n+1 CPn.. Submanifolds .. Smooth maps of maximal rank .. rank of a smooth map .. diffeomorphisms .. sets, submersions.
Chapter 1 Introduction 1.1 Some history In the words of S.S. Chern, ”the fundamental objects of study in differential geome-try are manifolds.” 1 Roughly, an n-dimensional manifold is a mathematical object that “locally” looks like Rn.The theory of manifolds has a …
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