微分形式及其應(yīng)用

前言

　　This is a free translation of a set of notes published originally in Portuguese in1971. They were translated for a course in the College of Differential Geometry， ICTP， Trieste， 1989. In the English translation we omitted a chapter onthe Frobenius theorem and an appendix on the nonexistence of a completehyperbolic plane in euclidean 3-space （Hilberts theorem）. For the presentedition， we introduced a chapter on line integrals.　　In Chapter 1 we introduce the differential forms in Rn. We only assumean elementary knowledge of calculus， and the chapter can be used as a basisfor a course on differential forms for "users" of Mathematics.　　In Chapter 2 we start integrating differential forms of degree one alongcurves 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. Itis useful （but not essential） that the reader be familiar with the notion of aregular surface in R3.　　In Chapter 4 we introduce the notion of manifold with boundary andprove Stokes theorem and Poincares lemma.　　Starting from this basic material， we could follow any of the possi-ble routes for applications： Topology， Differential Geometry， Mechanics， LieGroups， etc. We have chosen Differential Geometry. For simplicity， we restricted ourselves to surfaces.　　Thus in Chapter 5 we develop the method of moving frames of Elie Cartanfor surfaces. We first treat immersed surfaces and next the intrinsic geometryof surfaces　　Finally， in Chapter 6， we prove the Gauss-Bonnet theorem for compactorientable surfaces. The proof we present here is essentially due to S.S.Chern.We also prove a relation， due to M. Morse， between the Euler characteristicof such a surface and the critical points of a certain class of differentiablefunctions on the surface.

內(nèi)容概要

本書是一部簡短的微分幾何教程。詳細(xì)講述了微分幾何，并運(yùn)用它們研究曲面微分幾何的局部和全局知識。引入微分幾何的方式簡潔易懂，使得這本書非常適合數(shù)學(xué)愛好者。微分流形的介紹簡明，具體，以致最主要定理Stokes定理很自然得呈現(xiàn)出來。大量的應(yīng)用實(shí)例，如用E. Cartan的活動標(biāo)架方法來研究R3中浸入曲面的局部微分幾何以及曲面的內(nèi)蘊(yùn)幾何。最后一章集中所有來講述緊曲面Gauss-Bonnet定理的Chern證明。每章末都附有練習(xí)。目次：Rn中的微分幾何；線性代數(shù)；微分流形；流形上的積分；曲面的微分幾何；Gauss-Bonnet定理和Morse定理。

書籍目錄

Preface 1.Differential Forms in Rn 2.Line Integrals 3.Differentiable Manifolds 4.Integration on Manifolds; Stokes Theorem and Poincare's Lemma   1.Integration of Differential Forms   2.Stokes Theorem   3.Poincare's Lemma 5.Differential Geometry of Surfaces   1.The Structure Equations of R   2.Surfaces in R3   3.Intrinsic Geometry of Surfaces 6.The Theorem of Gauss-Bonnet and the Theorem of Morse   1.The Theorem of Gauss-Bonnet   2.The Theorem of Morse References Index

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