文档介绍：In tro duction and the pro of
The A tiy ah
P ato di
Singer index theorem
APS theorem
is used in this
text as a piv ot
or ma yb e an excuse
to discuss some asp ects of geometry
and analysis on manifolds with b oundary
Thisv olume do es not con tain a
general treatmen t of index theorems ev en though they are amongst the most
basic analytic
geometric results one can
nd
The p o w er of suc h theorems
in applications largely lies in their simplicit y and generalit y
In particular
the statemen t of the APS theorem is quite simple
In practice a great deal of
e
ort
b yman y p eople
has gone in to simplifying the pro ofs
This has lead
to
and b een panied b y
a m uc h wider understanding of the analytic
framew ork in whic h they are cen tred
In fact
from an analytic p ersp ectiv e
index theorems can b e though tofasm uc h as testing grounds
for metho ds
and concepts
as ends in themselv es
The A tiy ah
Singer theorem
whic his
the b oundaryless precursor to the APS theorem
is in timately connected to
the theory of pseudo di
eren tial op erators
This v olume is in tended to place
the APS theorem in a similar con text
the
b
category and related calculus
of b
pseudodifferen tial op erators on pact manifold with b oundary
The basic approac h adopted here is to
state
and
pro v e
the APS theo
rem immediately
b eing necessarily sup er
cial on a v ariet yofpoin ts
The
subsequen t nine c hapters consist largely in the
eshing out of this pro of
Just as the initial discussion is brief
the later treatmen t is discursiv eand
aims at considerably more than the pro of of the index theorem alone
The
pro of giv en here is dir e ct in t w o senses
The written pro of itself is quite
straigh tforw ard
giv en some conceptual bac kground
and in particular the
terms in the
nal form e out directly in the course of the pro of
The
mo del here is Getzler
s pro of
of the A tiy ah
Singer theorem for Dirac
op erators on pac