文档介绍：INV ARIANCE THEOR Y THE HEA T EQUA TION AND THE A TIY AH SINGER INDEX THEOREM b y P eter B Gilk ey Electronic reprin t cop yrigh t P eter B Gilk ey Bo ok originally published on pap er b y Publish or P erish Inc USA Library of Congress Catalog Card Num b er ISBN INTR ODUCTION This book treats the A tiy ah Singer index theorem using heat equation metho ds The heat equation giv es a lo cal form ula for the index of an y plex W e use in v ariance theory to iden tify the in tegrand of the index theorem for the four classical plexes with the in v ari an ts of the heat equation Since the t wisted plex pro vides a su cien tly ric h family of examples this approac h yields a pro of of the A tiy ah Singer theorem plete generalit y W e also use heat equation metho ds to discuss Lefsc hetz xed poin t form ulas the Gauss the orem for a manifold with smo oth b oundary and the t wisted eta in v arian t W e shall not include a discussion of the signature theorem for manifolds with b oundary The rst c hapter reviews results from analysis Sections through represen t standard elliptic material Sections through con tain the material necessary to discuss Lefsc hetz xed p oin t form ulas and other top ics In v ariance theory and di eren tial geometry pro vide the necessary link b e t w een the analytic form ulation of the index theorem giv en b y heat equation metho ds and the top ological form ulation of the index theorem con tained in the A tiy ah Singer theorem Sections through are a review of c har acteristic classes from the poin t of view of di eren tial forms Section giv es an in v arian t theoretic c haracterization of the Euler form whic hisused to giv e a heat equation pro of of the Gauss theorem Sections and discuss the P on trjagin forms of the tangen t bundle and the Chern forms of the co e cien t bundle using in v ariance theory