指标定理概述.pdf

摘要摘要 Atiyah-Singer指标定理是数学宝库的一颗璀璨的明珠,它在微分几何,偏微分方程,算子代数以及理论物理等众多领域建立联系。、拓扑学以及代数几何学中的诸多大定理。指标定理是在不断的发展和推广中的,(文献1)(文献5,6)(文献 10)。整体指标定理给出了椭圆算子的Fredholm指标解释,作为特殊的椭圆算子Dirac算子,它的指标是局部得到的,而Dirac算子的指标形式被称为局部指标定理。我国著名数学家虞言林教授在八十年代给出了局部指标定理的一个证明。本文中第一章介绍指标定理,第二章引入示性类和重要的椭圆微分算子等基础知识,第三章介绍高斯一博内特一陈省身(Gauss..Chern)定理、希策布鲁赫(Hirzebruch)符号差定理、希策布鲁赫一黎曼一洛赫()定理,第四章简单介绍指标定理的热方程证明方法。关键词:指标定理椭圆微分算子示性类热方程 Abstract Atiyah-Singer Index theorem is abrightpearl treasure house establishes contact、订thmany fieldssuch differential equations,operator algebras andthetheory ’S SO beautiful andplayan important roleinmathematics,For example,it contains thedifferentialgeometry, topology and algebraic geometry inmany big theorems. Index theorem isinthecontinued development ’S proof method isdivided into thefollowing theory(1iteraturel)2. K—theory(1iterature 5,6).(1iterature0) Indextheorem gives theinterpretation oftheFredholm index ofellipticoperators. As aspecial ellipticoperator Dirac operator,it islocally obtained index,index formof the Dirac operator iscalledthelocalindex famous naathematician, Professor Yu Yanlin give aproof ofthelocalindex theorem in 980. The firstchapter inthispaper introduces second chapter introduced thecharacteristic thirdchapter introduces Gauss—---Chem theory,Hirzebruch theory,Hirzebruch-Riemann--Roeh theory. Keywords:Index theorem elliptic differentialoperator Characteristic classes heatequation 目录目录摘要???????????????????????????.I Abstract..?..?..?.?...?..?.??.?..??...?.?..?.??.?.?.?..?..?...?...II 第一章指标定理的概述?????????????????1 第二章椭圆微分算子和示性类?????????????..3 第一节椭圆微分算子????????????????一3 第二节示性类????????????????????4 第三章指标定理的三个特例??????????????11 第一节(Gauss—.Chem)定理???????????11 第二节(Hirzebruch)符号差定理????????????..12 第三节(Hirzebruch—Rieman