Differential Forms and Applications

  • Manfredo P. do Carmo

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages I-IX
  2. Manfredo P. do Carmo
    Pages 1-16
  3. Manfredo P. do Carmo
    Pages 17-31
  4. Manfredo P. do Carmo
    Pages 33-54
  5. Manfredo P. do Carmo
    Pages 77-98
  6. Manfredo P. do Carmo
    Pages 99-113
  7. Back Matter
    Pages 115-120

About this book

Introduction

This is a free translation of a set of notes published originally in Portuguese in 1971. They were translated for a course in the College of Differential Geome­ try, ICTP, Trieste, 1989. In the English translation we omitted a chapter on the Frobenius theorem and an appendix on the nonexistence of a complete hyperbolic plane in euclidean 3-space (Hilbert's theorem). For the present edition, we introduced a chapter on line integrals. In Chapter 1 we introduce the differential forms in Rn. We only assume an elementary knowledge of calculus, and the chapter can be used as a basis for a course on differential forms for "users" of Mathematics. In Chapter 2 we start integrating differential forms of degree one along curves in Rn. This already allows some applications of the ideas of Chapter 1. This material is not used in the rest of the book. In Chapter 3 we present the basic notions of differentiable manifolds. It is useful (but not essential) that the reader be familiar with the notion of a regular surface in R3. In Chapter 4 we introduce the notion of manifold with boundary and prove Stokes theorem and Poincare's lemma. Starting from this basic material, we could follow any of the possi­ ble routes for applications: Topology, Differential Geometry, Mechanics, Lie Groups, etc. We have chosen Differential Geometry. For simplicity, we re­ stricted ourselves to surfaces.

Keywords

Diferential forms Differentialformen begleitendes Dreibein differential geometry differential geometry of surfaces differential manifolds differenzierbare Mannigfaltigkeit eingebettete Flächen immersed surfaces intrinsic geeometry of surfaces intrinsische Geometrie von Flächen manifold moving frames

Authors and affiliations

  • Manfredo P. do Carmo
    • 1
  1. 1.Instituto de Matematica Pura e Aplicada (IMPA)Rio de JaneiroBrazil

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-57951-6
  • Copyright Information Springer-Verlag Berlin Heidelberg 1994
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-540-57618-1
  • Online ISBN 978-3-642-57951-6
  • Series Print ISSN 0172-5939
  • Series Online ISSN 2191-6675
  • About this book