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Introduction to Geometry and Topology

  • Werner Ballmann

Part of the Compact Textbooks in Mathematics book series (CTM)

Table of contents

  1. Front Matter
    Pages i-x
  2. Werner Ballmann
    Pages 1-25
  3. Werner Ballmann
    Pages 27-67
  4. Werner Ballmann
    Pages 69-101
  5. Werner Ballmann
    Pages 103-152
  6. Back Matter
    Pages 153-169

About this book

Introduction

This book provides an introduction to topology, differential topology, and differential geometry. It is based on manuscripts refined through use in a variety of lecture courses. The first chapter covers elementary results and concepts from point-set topology. An exception is the Jordan Curve Theorem, which is proved for polygonal paths and is intended to give students a first glimpse into the nature of deeper topological problems.

The second chapter of the book introduces manifolds and Lie groups, and examines a wide assortment of examples. Further discussion explores tangent bundles, vector bundles, differentials, vector fields, and Lie brackets of vector fields. This discussion is deepened and expanded in the third chapter, which introduces the de Rham cohomology and the oriented integral and gives proofs of the Brouwer Fixed-Point Theorem, the Jordan-Brouwer Separation Theorem, and Stokes's integral formula.

The fourth and final chapter is devoted to the fundamentals of differential geometry and traces the development of ideas from curves to submanifolds of Euclidean spaces. Along the way, the book discusses connections and curvature--the central concepts of differential geometry. The discussion culminates with the Gauß equations and the version of Gauß's theorema egregium for submanifolds of arbitrary dimension and codimension.

This book is primarily aimed at advanced undergraduates in mathematics and physics and is intended as the template for a one- or two-semester bachelor's course.

Keywords

field cohomology curvature curv Liesche Group plurality context

Authors and affiliations

  • Werner Ballmann
    • 1
  1. 1.Max-Planck-Institut für MathematikBonnGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-0348-0983-2
  • Copyright Information Springer Basel 2018
  • Publisher Name Birkhäuser, Basel
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-0348-0982-5
  • Online ISBN 978-3-0348-0983-2
  • Series Print ISSN 2296-4568
  • Series Online ISSN 2296-455X
  • Buy this book on publisher's site