文档介绍：INV ARIANCE THEOR Y
THE HEA T EQUA TION
THE
AND
A TIY AH
SINGER
INDEX THEOREM
b y P eter B
Gilk ey
Electronic reprin t
cop yrigh t
P eter B
Gilk ey
published on pap er b y Publish or P erish Inc
USA
Bo ok originally
Library of Congress Catalog Card Num b er
ISBN
INTR ODUCTION
b o ok treats the A tiy ah
Singer index theorem using heat equation
This
metho ds
The heat equation giv es a lo cal form ula for the index of an y
elliptic
complex
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
tly ric h family of examples
this approac h yields a pro of of the
a su
cien
A tiy ah
Singer theorem plete generalit y
W e also use heat equation
hetz
xed p oin t form ulas
the Gauss
the
metho ds to discuss Lefsc
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
through
The
rst c hapter reviews results from analysis
Sections
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
through
are a review of c har
the A tiy ah
Singer theorem
Sections
acteristic classes from the p oin t of view of di
eren tial forms
Section
giv es an in v arian t
theoretic c haracterization of the Euler form whic h is used
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
cie