Differential Forms and Applications (Universitext)

The Book Treats Differential Forms And Uses Them To Study Some Local And Global Aspects Of The Differential Geometry Of Surfaces. Differential Forms Are Introduced In A Simple Way That Will Make Them Attractive To Users Of Mathematics. A Brief And Elementary Introduction To Differentiable Manifolds Is Given So That The Main Theorem, Namely The Stokes' Theorem, Can Be Presented In Its Natural Setting. The Applications Consist In Developing The Method Of Moving Frames Of E. Cartan To Study The Local Differential Geometry Of Immersed Surfaces In R3 As Well As The Intrinsic Geometry Of Surfaces. Everything Is Then Put Together In The Last Chapter To Present Chern's Proof Of The Gauss-bonnet Theorem For Compact Surfaces. Differential Forms In R(symbol) -- Line Integrals -- Differentiable Manifolds -- Integration On Manifolds; Stokes Theorem And Poincare's Lemma -- Differential Geometry Of Surfaces -- The Theorem Of Gauss-bonnet And The Theorem Of Morse. Manfredo P. Do Carmo. Includes Bibliographical References (p. [115]) And Index. 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 The book treats differential forms and uses them to study some local and global aspects of the differential geometry of surfaces. Differential forms are introduced in a simple way that will make them attractive to "users" of mathematics. A brief and elementary introduction to differentiable manifolds is given so that the main theorem of differential forms, namely Stokes' theorem, can be presented in its natural setting. The applications consist in developing the method of moving frames of E. Cartan to study the local differential geometry of immersed surfaces in R[superscript 3] as well as the intrinsic geometry of surfaces. Everything is then put together in the last chapter to present Chern's proof of the Gauss-Bonnet theorem for compact surfaces.

The book treats differential forms and uses them to study some local and global aspects of the differential geometry of surfaces. Differential forms are introduced in a simple way that will make them attractive to "users" of mathematics. A brief and elementary introduction to differentiable manifolds is given so that the main theorem, namely the Stokes' theorem, can be presented in its natural setting. The applications consist in developing the method of moving frames of E. Cartan to study the local differential geometry of immersed surfaces in R3 as well as the intrinsic geometry of surfaces. Everything is then put together in the last chapter to present Chern's proof of the Gauss-Bonnet theorem for compact surfaces.

An application of differential forms for the study of some local and global aspects of the differential geometry of surfaces. Differential forms are introduced in a simple way that will make them attractive to "users" of mathematics. A brief and elementary introduction to differentiable manifolds is given so that the main theorem, namely Stokes' theorem, can be presented in its natural setting. The applications consist in developing the method of moving frames expounded by E. Cartan to study the local differential geometry of immersed surfaces in R3 as well as the intrinsic geometry of surfaces. This is then collated in the last chapter to present Chern's proof of the Gauss-Bonnet theorem for compact surfaces. Front Cover......Page 1 Title Page......Page 4 Copyright Information......Page 5 Dedication......Page 6 Preface......Page 8 Contents......Page 10 1. Differential Forms in R^n......Page 12 2. Line Integrals......Page 28 3. Differentiable Manifolds......Page 44 1. Integration of Differential Forms......Page 66 2. Stokes Theorem......Page 71 3. Poincaré's Lemma......Page 77 1. The Structure Equations of R^n......Page 88 2. Surfaces in R^3......Page 93 3. Intrinsic Geometry of Surfaces......Page 100 1. The Theorem of Gauss-Bonnet......Page 110 2. The Theorem of Morse......Page 117 References......Page 126 Index......Page 128 Back Cover......Page 133 This title by M. do Carmo, winner of the 1992 Mathematics Price of the Third World Academy of Sciences, gives an introduction to the theory of differentiable forms. Since it only assumes elementary calculus and elementary linear algebra, it is suitable for second year undergraduate and graduate students in mathematics and physics. The goal of this chapter is to define in Rn "fields of alternate forms" that will be used later to obtain geometric results.