黎曼幾何概論

內(nèi)容概要

　　《黎曼幾何概論》既是一本學(xué)習(xí)黎曼幾何發(fā)展參考書，也是一本很好的教程，包括了學(xué)習(xí)現(xiàn)代微分幾何研究生需要了解的方方面面。黎曼幾何變得越來越重要，《黎曼幾何概論》中著手本領(lǐng)域比較熟悉的話題，并且盡快過渡到最新科研成果。這些結(jié)果并沒有給出詳細(xì)的證明，但一些的重要的結(jié)果仍然描述的十分詳細(xì)生動，使得讀者對該領(lǐng)域有詳細(xì)深刻的理解。然而，黎曼流形作為一個很小的科目，概念的建立需要一個過程，前三章側(cè)重于通過常規(guī)的方法引入各種概念和黎曼幾何的工具，緊接著詳細(xì)講述gauss和riemann。

作者簡介

作者：（法國）貝格（Marcel Berger）

書籍目錄

1　Euclidean Geometry
　1.1　Preliminaries
　1.2　Distance Geometry
　　1.2.1 A Basic Formula
　　1.2.2 The Length of a Path
　　1.2.3 The First Variation Formulaand Application to
Billiards
　1.3　Plane Curves
　　1.3.1 Length
　　1.3.2 Curvature
　1.4　Global Theory of Closed Plane Curves
　　 1.4.1 "Obvious" Truths About Curves Which are Hard
to Prove
　　 1.4.2 The Four Vertex Theorem
　　 1.4.3 Convexity with Respect to Arc Length
　　 1.4.4 Umlaufsatz with Corners
　　 1.4.5 Heat Shrinking of Plane Curves
　　 1.4.6 ArnoFd's Revolution in Plane Curve
Theory
　1.5　The Isoperimetric Inequality for Curves
　1.6　The Geometry of Surfaces Before and After Gaufl
　　 1.6.1 Inner Geometry:. a First Attempt
　　 1.6.2 Looking for Shortest Curves: Geodesics
　　 1.6.3 The Second Fundamental Formand Principal
Curvatures
　　 1.6.4 The Meaning of the Sign of K
　　 1.6.5 Global Surface Geometry
　　 1.6.6 Minimal Surfaces
　　 1.6.7 The Hartman-Nirenberg Theoremfor Inner Flat
Surfaces
　　 1.6.8 The Isoperimetric Inequality in E3 ~ la
Gromov
　　1.6.8.1 Notes
　1.7　Generic Surfaces
　1.8　Heat and Wave Analysis in
　　 1.8.1 Planar Physics
　　1.8.1.1 Bibliographical Note
　　……

章節(jié)摘錄

版權(quán)頁：   插圖：   This would describe the motion of the sea,if there were nocontinents on the planet.It might not be a bad approximation,since oceansare the bulk of the Earth's surface.His first result backs intuition.Disturb the water with a big shock（a Dirac distribution,if you prefer）somewhere,say at the north pole.Then you will always find another big shock at the south pole after a time T which is no larger than 2π.But Meyer's second series of results oppose intuition: the big shock can move from the north to the south pole while remaining extremely small everywhere for all time between O and T,as in figure 1.102 on the next page.Worse some apparently moderate shocks can,with larger and larger time,be very small almost everywhere and almost all the time,but at some times and some places can be as big as desired（everything there and then will break down）.Meyer's results explained the following strange observation: quite recently a tidal wave was observed at Martinique,and came back there later,this being completely unnoticed eyerywhere else in between times. 1.9.3 Billiards in Higher Dimensions  The Faber-Krahn inequality extends to any dimension.Again,the proof involves the isoperimetric inequality（in any dimension）.Contrarily,billiards in dimensions larger than two are still to be fully explored.There is one ex-ception: billiards in concave regions,for which we refer to Katok,Strelcyn,Ledrappier & Przytycki 1986（788）.Such regions have the best possible ergodic behavior.But for nonconcave regions,even for polyhedra in three dimensional space,almost nothing is known concerning periodic motions or ergodicity,except for rectangular parallelepipeds,balls（of course）and the Berard examples mentioned above.There is an unpublished result of Katok asserting that the length counting function is subexponential.But,even in the simplest possi-ble cases,things are either not finished,or completely open.Let us mention only two cases.The first is that of the cube.Dynamics in a cube are compa-rable to dynamics in a square: a trajectory is periodic or everywhere dense.

編輯推薦

《黎曼幾何概論》既是一本學(xué)習(xí)黎曼幾何發(fā)展參考書，也是一本很好的教程，包括了學(xué)習(xí)現(xiàn)代微分幾何研究生需要了解的方方面面。黎曼幾何變得越來越重要，《黎曼幾何概論》中著手本領(lǐng)域比較熟悉的話題，并且盡快過渡到最新科研成果。這些結(jié)果并沒有給出詳細(xì)的證明，但一些的重要的結(jié)果仍然描述的十分詳細(xì)生動，使得讀者對該領(lǐng)域有詳細(xì)深刻的理解。

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