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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

1. Differential Forms in R n

Abstract
The goal of this chapter is to define in Rn “fields of alternate forms” that will be used later to obtain geometric results.
Manfredo P. do Carmo

2. Line Integrals

Abstract
Differential forms are to be integrated. We will do that soon (Chapter 4) after some preliminaries on the natural “habitat” of differential forms (Chapter 3). However, the special case of integration of forms of degree one along curves (the so called line integrals) is so simple that it can be treated independently of the general theory. We will do that in this chapter.
Manfredo P. do Carmo

3. Differentiable Manifolds

Abstract
Differential forms were introduced in the first chapter as objects in Rn; however, they, as everything else that refers to differentiability, live naturally in a differentiable manifold, a concept that we will develop presently.
Manfredo P. do Carmo

4. Integration on Manifolds; Stokes Theorem and Poincaré’s Lemma

Abstract
In this section we will define the integral of a differential n-form on an n- dimensional differentiable manifold. We will start with the case of Rn.
Manfredo P. do Carmo

5. Differential Geometry of Surfaces

Abstract
We now apply our knowledge of differential forms to study some differential geometry. We start with a few definitions.
Manfredo P. do Carmo

6. The Theorem of Gauss-Bonnet and the Theorem of Morse

Abstract
The considerations of the last chapter were strictly local. However, one of the most interesting features of differential geometry is the connection between local properties and properties that depend on the entire surface. One of the most striking of such properties is the so-called Gauss-Bonnet theorem which we intend to prove in this section.
Manfredo P. do Carmo

Backmatter

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