Lee K.K. Lectures on Dynamical Systems, Structural Stability and Their Applications
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If
we
define
llfll =
g.l.b.
(kl
llf(x)
II
S kllxll
for
all
x e
B
1
},
then
this
norm
makes
the
space
of
linear
maps
of
Banach
spaces,
into
a
Banach
space
Lin(B
1
,B
2
).
Let
u
1
~;;
B
1
,
U
2
~;;
B
2
are
open
subsets
of
Banach
spaces,
and
let
g:
u
1
~
u
2
be
continuous,
g
is
differentiable
at
x e
u
1
with
derivative
g'
E
Lin(B
1
,B
2
)
if
for
all
h e B
1
of
sufficiently
small
norm
e(h)
=
g(x
+h)
-
g(x)
-
g'(x)h
satisfies
limlhl~o
II
E
(h)/llhll
II
= o.
It
is
not
difficult
to
check
the
usual
linearity
and
the
chain
rule.
Next
we
shall
introduce
the
notion
of
an
infinite
dimensional
manifold
modeled
after
some
given
topological
vector
space,
rather
than
just
usual
R",
and
discuss
some
basic
properties.
Let
V
be
a
topological
vector
space
(usually
a
Banach
space).
A T
2
space
Miscalled
a
manifold
modeled
on
V,
if
for
every
p e M
there
is
some
open
subset
OP
of
M,
with
p e
OP,
and
a
homeomorphism
¢p
of
OP
onto
an
open
subset
of
v.
Of
course
there
are
many
OP
and
¢p
for
each
p e
M.
This
definition
is
very
similar
to
an
earlier
definition
of
a
finite
dimensional
manifold
except
that
¢p
is
only
required
to
be
a
homeomorphism
onto
an
open
subset
of
V,
not
onto
all
of
V.
In
fact,
in
general
we
shall
not
be
able
to
specify
the
range
of
¢p·
It
is
clear
that
every
finite
dimensional
manifold
is
a
manifold
modeled
on
R"
for
some
n.
If
V
is
a
topological
vector
space,
then
any
open
subset
of
V
is
a
manifold
modeled
on
v.
If
M"
is
a
finite
dimensional
manifold
and
B a
Banach
space,
then
M"xB
is
a
manifold
modeled
on
the
(Banach)
space
R"xB.
It
is
more
convenient
to
work
with
norms
in
order
to
define
an
infinite
dimensional
manifold.
For
the
time
being,
we
shall
restrict
our
attention
to
manifolds
modeled
on
a
Banach
space
and
it
will
be
called
a
Banach
manifold.
Let
B
be
a
given
Banach
space,
M
is
a
smooth
manifold
modeled
on
B
if
M
is
a
manifold
modeled
on
B
such
that
if
OP
n
Oq
+ o,
th~n
the
composite
map
¢
(0
n o )
c•
1
•fco,n
«1>>
....
o n o
•c-+
¢
(O
n o )
pp
q p q
qp
q
is
a
smooth
map
in
the
sense
that
it
has
continuous
124
derivatives
of
any
order
(obtained
from
iterating
the
definition
of
the
differentiable
at
a
given
point).
Let
M
and
N
are
manifolds
modeled
on
Banach
spaces.
Let
f:M
~
N
be
a
continuous
map. f
is
smooth
Cor
differentiable)
if
for
each
p e
OP
~
M
with
homeomorphism
cpP:
OP
~
cpP(OP)
and
q e
oq
~
N
with
tPq:
oq
~
tPq(Oq),
where
f(p)
=
q,~
then
in
a
sufficiently
small
neighborhood
of
p,
t/Jp(OP)
•,'
~
M
f~
N
•r~
tPq(Oq)
is
smooth
(or
differentiable).
Some
of
the
notions
for
finite
dimensional
manifolds
can
easily
be
generalized
to
our
current
interest,
nonetheless,
occasionally
some
care
has
to
be
exercised.
Submanifold
is
defined
by
requiring
that
each
point
has
a
neighborhood
homeomorphic
to
0
1
xo
2
where
0
1
and
o
2
are
open
in
B
1
and
B
2
respectively.
And
the
big
manifold
is
modeled
on
B
1
xB
2
,
and
the
manifold
is
described
locally
in
terms
of
o
1
x{pt}.
One
can
easily
check
that
such
a
submanifold
is
a
manifold.
A
diffeomorphism
of
infinite
dimensional
manifolds
is
a
smooth
map
with
a
smooth,
two
sided
inverse.
An
imbedding
cp:
M
~
N
is
a
smooth
one-to-one
map
that
is
a
diffeomorphism
of
M
onto
a
submanifold
of
N.
An
immersion
is
a
smooth
map
which
is
locally
an
imbedding.
For
a
given
smooth
manifold
M
modeled
on
a
Banach
space
B,
let
us
choose
a
coordinate
neighborhood
oP,
for
each
p e
M,
endowed
with
a
homeomorphic
cpP
onto
an
open
set
in
B,
i.e.,
cpP:
OP
~
t/Jp(OP)
~
B.
Consider
the
sets
OPxB
and
define
an
equivalence
relation
in
their
union
by
specifying
that
if
(u,v
1
)
e
OPxB,
(u,v
2
)
e
OqxB,
then
(u,v
1
)
is
equivalent
to
(u,v
2
)
iff
(cpq
·
cpp·
1
)
1
(v
1
)
= v
2
where
the
prime
refers
to
the
derivative
taken
at
u.
The
quotient
space
is
defined
to
be
the
tangent
bundle
TM
and
the
projection
map " :
TM
~
M
is
defined
by
"{(u,v)}
=
u.
And
as
before,
the
fiber
is
the
tangent
space
at
p,
i.e.,
""
1
(p)
=
MP.
As
before,
the
differential
of
a
smooth
map
between
smooth
manifolds
can
be
defined.
Then
it
is
possible
to
characterize
immersion
in
terms
of
the
tangent
space
by
requiring
that
at
every
point
the
differentials
of
the
125
immersion
(df)P:
MP
-+
Nf<p>
have
a
left
inverse.
By
generalizing
the
inverse
function
theorem
to
Banach
spaces
one
can
prove
the
equivalence
of
the
two
notions
of
immersion.
Since
a
separable
Hilbert
manifold,
a
manifold
modeled
on
L
2
,
is
a
fortiori
a
paracompact
space,
hence
partition
of
unity
exists
from
a
theorem
in
point-set
topology.
It
can
be
shown
that
such
a
manifold
has
a
smooth
partition
of
unity
[Kahn
1980,
p.216].
Theorem
3.4.1
Let
M
be
a
separable
manifold
modeled
on
L
2
,
and
let
{Ba}
be
an
open
covering,
where
Ba(x)
is
an
open
ball
of
radius
a > 0
about
x f L
2
•
Then
there
is
a
countable,
locally
finite
open
covering
{On}, a
refinement
of
{Ba},
and
there
exists
a
smooth
partition
of
unity
subordinate
to
this
covering
{On}·
Let
M
be
a
smooth
m-dim
Riemannian
manifold,
i.e.,
for
any
p f
M,
and
vectors
u,
v f
MP
we
have
a
continuous,
symmetric,
positive
definite
inner
product
defined
by
(u,v)P
=
ut(g(p)
)v
and
the
norm
is
defined
by
llullp =
~
(u,u)P.
Let
X
be
any
compact
Hausdorff
space,
M
be
a
Riemannian
manifold.
We
set
F(X,M) =
{fl
f:
X-+
M,
f
continuous}.
A
metric
on
F(X,M)
can
be
introduced
d(f,g)
=
l.u.b.x•X
a(f(x)
,g(x)),
a(a,b)
is
the
greatest
lower
bound
smooth
curves
in
M
from
a
to
b.
It
makes
F(X,M)
into
a
metric
space.
by
f,g
f F(X,M)
where
of
the
length
of
all
is
easy
to
show
that
d
For
a
given
f f
F(X,M),
one
may
define
a
tangent
space
to
F(X,M)
at
that
given
map,
denoted
by
F(X,M)f,
by
letting
¢:
TM
-+
M
be
the
projection
map
of
the
tangent
bundle
of
M.
Then
F(X,M)f
is
the
set
of
all
f':
X-+
TM
such
that
~·f'
f.
Obviously
one
can
make
F(X,M)f
into
a
linear
space
because
if
we
set
(f
1
'
+ f
2
')
(x)
= f
1
'
(x)
+ f
2
'
(x),
where
f
1
',
t
2
•:
X-+TMandx
f
X,
then~·(f
1
'
+ f
2
')(x)
=
f(x)
and
likewise
for
af'.
One
can
also
introduce
a
norm
into
F(X,M)f
by
setting
llf'llt
=
l.u.b.x•X
lit'
(x)
llf<x>"
Note
that
lit'
(x)
llf<x>
is
the
norm
in
Mf<x>
defined
by
the
Riemannian
metric.
It
is
then
straight
forward
to
show
that
F(X,M)f
is
a
complete
126
metric
space
with
respect
to
the
norm.
That
is,
F(X,M)f
is
a
real
Banach
space.
Theorem
3.4.2
If
X
is
a
compact
Hausdorff
space
and
M
is
a
smooth
Riemannian
manifold,
then
F(X,M)
is
a
smooth
manifold
modeled
on
a
real
Banach
space
-
F(X,M)f,
independent
of
the
choice
of
f
[Eells
1958,
see
also
Eells
1966,
Kahn
1980,
p.218].
If
here
X
and
M
are
compact
manifolds,
as
before,
F(X,M)f
be
the
vector
space
of
all
smooth
liftings
of
f e
F(X,M),
i.e.,
f'
e
F(X,M)f
is
smooth
and
f':
X~
TM
such
that
~·f'
= f
where~=
TM
~
M
is
the
canonical
projection.
Earlier,
we
have
pointed
out
that
such
space
F(X,M)f
is
a
complete
linear
vector
space,
thus
a
Frechet
space.
(A
Frechet
space
is
a
topological
vector
space
which
is
metrizable
and
complete).
Thus
it
is
not
surprising
to
note
that:
Theorem
3.4.3
The
group
of
diffeomorphisms
of
M,
Diff(M),
is
a
locally
Frechet
~group
[Leslie
1967].
For
the
relations
between
the
homeomorphism
group
and
the
diffeomorphism
group
of
a
smooth
manifold
and
their
homotopic
type,
one
should
consult
[Burghelea
and
Lashof
1974a,b].
In
addition
to
the
manifold
and
differentiable
structures
for
spaces
of
differentiable
maps,
one
can
also
construct
such
structures
to
a
more
general
class,
the
spaces
of
sections
of
fiber
bundles.
Let
a
be
a
smooth
vector
bundle
over
a
compact
smooth
manifold
M,
then
define
S(a)
to
be
the
set
of
all
sections
s
of
a
such
that
s E
S(~)
for
some
open
vector
subbundle
~
of
a,
i.e.,
S(a)
=
U~
S(~)
where
the
union
is
over
all
open
vector
subbundles
~
of
a.
Then
one
can
show
that
S(a)
not
only
is
a
Banach
manifold,
but
also
has
a
unique
differentiable
structure
[see
Palais
1968].
We
have
only
scratched
the
surface
of
this
evolving
and
very
interesting
area.
Interested
readers
are
urged
to
consult
Lang
[1962],
Eells
[1966],
Burghelea
and
Kuiper
[1969],
Eells
[1958],
Eells
and
Elworthy
[1970],
Marsden,
127
Ebin,
and
Fischer
(1972],
and
some
papers
in
Anderson
(1972],
Palais
(1965,
1966a,b,
1971].
Although
most
results
in
finite
dimensional
manifolds
can
easily
be
extended
to
infinite
dimensional
manifolds,
nonetheless,
there
are
a
few
surprises.
It
has
been
established
that
every
separable,
metrizable
C
0
-manifold
can
be
C
0
-imbedded
as
an
subset
of
its
model
(Henderson
1969].
In
other
words,
any
reasonable
Hilbert
manifold
is
equivalent
to
an
open
subset
of
L
2
space.
This
is
contrary
to
the
case
of
finite
dimensional
manifold!
Let
M
be
a
~
manfold
modeled
on
any
infinite
dimensional
Hilbert
space
E,
Kuiper
(1965]
asserts
that
M
is
parallelizable.
Eells
and
Elworthy
(1970]
observe
that
there
is
a
diffeomorphism
of
M
onto
MxE.
They
also
observe
that
if
M
and
N
are
two
~
manifolds
modeled
on
E
and
if
there
is
a
homotopy
equivalence
~:
M
~
N,
then
~
is
homotopic
to
a
diffeomorphism
of
M
onto
N.
As
a
corollary,
the
differentiable
structure
on
M
is
unique.
Again,
these
results
are
different
from
the
finite
dimensional
cases
as
we
have
pointed
out
earlier
in
Sect.2.4
(see
also
Milnor
1956].
3.5
Differential
operators
Most
of
the
basics
of
linear
differential
operators
may
be
generalized
to
manifolds
by
piecing
together
differential
operators
on
Euclidean
spaces,
but
it
turns
out
much
more
elegant
and
convenient
to
give
a
general
definition
in
terms
of
vector
bundles.
We
shall
adopt
this
approach
to
avoid
cumbersome
details
involving
coordinate
charts
at
the
beginning.
After
that,
we
shall
discuss
various
important
cases
and
examples
of
differential
operators.
Finally,
we
shall
briefly
discuss
the
important
notions
of
ellipticity,
the
symbol
of
an
operator,
linearization
of
nonlinear
operators,
and
the
analytic
index
of
operators.
References
for
further
reading
will
be
provided.
Let
(B
1
,M,~
1
)
and
(B
2
,M,~
2
)
be
two
smooth
vector
bundles
over
an
m-dim
compact
smooth
manifold
M,
and
let
Cm(M,B;)
be
128
the
vector
space
of
sections
of
the
bundle
(Bi,M'~i),
here
i
=
1,
2.
A
linear
differential
operator
is
a
linear
map
of
vector
spaces
P:
c"'(M,B
1
)
....
c"'(M,B
2
)
such
that
supp
P(s)
s:
supp
s,
where
supp
s
=closure
of
{p
£ M I
s(p)
+
0}.
Note
that
this
definition
is
very
elegant
and
simple,
nonetheless,
some
work
is
needed
so
that
we
can
"visualize"
that
these
operators
are
locally
generated
by
differentiation.
In
the
following,
we
shall
specify
the
smooth
manifold
M
and
its
vector
bundles,
the
linear
map,
and
sections
to
make
the
definition
"visualizable".
If
M =
Rm
and
B
1
,
B
2
are
trivial
1-dim
vector
bundles
(line
bundles)
over
M,
and
if
a=
(a
1
,
•••
,am)
is
an
index
set,
then
for
a
smooth
f
one
can
define
a
linear
differential
operator
D"
by
D"f =
alalf;ax,
0
•
•••
a~a
....
If
P
is
any
polynomial
over
the
ring
c"'(M,R)
in
m
variables
z,,
•••
,zm,
then
if
we
substitute
a;axi
for
zi,
then
the
resulting
polynomial
gives
a
linear
differential
operator
P
:
c"'(M,R)-+
c"'(M,R)
by
P(f)
=
P(a;ax
1
,
•••
, a;axm)
(f).
If
s
vanishes
in
an
open
set,
every
term
of
P(s)
and
P(s)
itself
vanishes
on
that
open
set.
Clearly
supp
P(s)
s:
supp
s.
Let
B
1
,
B
2
and
M
be
as
before,
and
let
P:
c"'(M,B
1
)
....
c"'(M,B
2
)
be
a
linear
differential
operator.
Then
we
say
that
P
has
order
k
at
the
point
p £ M
if
k
is
the
largest
non-negative
integer
such
that
there
is
some s £ c"'(M,B
1
)
and
some
smooth
function
f
defined
in
an
open
neighborhood
of
p
and
vanishing
at
p
such
that
P(fks)
(p)
+ o.
The
order
of
P
is
the
maximum
of
the
orders
of
P
at
all
points
of
M.
It
is
easy
to
check
that
this
notion
of
order
for
P
agrees
with
the
usual
definition
of
order
for
a
linear
differential
operator
defined
on
Euclidean
space.
Here
a
word
of
caution
is
called
for.
Recall
in
our
definition
of
differential
operator
and
its
order,
we
assume
that
M
being
compact.
If
it
happens
that
M
is
noncompact,
then
the
order
of
a
linear
operator
may
not
exist.
For
example,
M = R
1
and
choose
rpi
£ c"'(R
1
)
= c"'(R
1
,R
1
)
with
support
supptJ>i
s:
[i,i+1]
and
f/>i(i+1/2) > o.
Set
P(f)(p)
=
Eit/>i(p)dif(p)/dxi.
Clearly,
with
such
construction,
P
is
a
linear
differential
operator,
129
but
the
order
of
P
is
not
defined.
Fortunately,
the
next
theorem
asserts
that
a
linear
differential
operator
locally
will
have
an
order,
and
consequently
a
linear
differential
operator
defined
on
a
compact
manifold
will
have
a
finite
order.
Tbeorem
3.5.1
(local
theorem)
Let
0
~
Rn
be
open
and
let
P:
c-co,t)
~
c-eo,~)
be
a
linear
differential
operator
from
the
sections
of
the
trivial
s-dim
vector
bundle
t
over
0
to
those
of
the
trivial
t-dim
vector
bundle
~
over
o.
Let
0
1
~
0
have
a
compact
closure
contained
in
o.
Let
V(t,~)
be
the
vector
space
of
linear
maps
from
R
5
to
Rt.
Then
there
is
an
m
~
0
such
that
for
every
multi-index
a,
lal
~
m,
there
are
c-
maps g
11
: 0
1
~
V(t,~)
such
that
for
any
f E
c-co,,t),
p
E 0
1
,
(Pf)
(p)
=
:EI
11
I:sm
g
11
(p)
(D
11
f)
(p)
[Kahn
1980,
p.194;
Peetre
1960).
The
next
theorem
has
the
essential
idea
and
it
is
a
global
theorem
for
differential
operators
defined
on
a
manifold.
Theorem
3.5.2
(global
theorem)
Let
M
be
a
smooth
manifold,
and
let
B
1
and
B
2
be
two
vector
bundles
over
M,
with
dim
B;
=
a;.
Let
P:
c-(M,B
1
)
~
c-(M,B
2
)
be
a
linear
differential
operator.
Take
p E
M,
then
there
is
a
coordinate
neighborhood
of
p,
u,
over
which
both
bundles
are
trivial
and
a
positive
integer
m
so
that
in
u,
(Pf)
(p)
=
:E
1
,.
1
:sm
g
11
(P)
(~f)
(p)
for
smooth
maps g
11
: U -+ V(B
1
,B
2
)
[Kahn
1980,
p.196].
In
the
last
section
we
have
mentioned
that
the
space
of
sections
of
a
vector
bundle
is
a
Banach
manifold
and
has
a
unique
differentiable
structure.
We
shall
say
a
little
bit
more
and
state
the
theorem
due
to
Hormander
[1964].
Let
(B,M,~)
be
a
smooth
vector
bundle,
s;
E c-(M,B)
be
a
sequence
of
c-
sections
of
B,
and
a
given
fixed
section
s.
We
say
that
~i
converges
to
s
locally
uniformly
if
for
any
p
E M
there
is
a
coordinate
neighborhood
U
of
p
over
which
B
is
trivial
such
that
in
U,
s;
and
~s;
converge
uniformly
to
s
and
~s,
respectively.
For
two
given
smooth
vector
bundles
B
1
and
B
2
,
a
linear
map L: c-(M,B
1
)
~
c-(M,B
2
)
is
weakly
130
continuous
if
whenever
si
converges
to
s
locally
uniformly,
then
L(si)
converges
to
L(s)
uniformly
over
K
~
M.
The
term
weakly
refers
to
the
fact
that
we
have
not
yet
topologized
the
vector
spaces
c-(M,Bi),
so
it
does
not
make
sense
to
ask
whether
L
is
continuous.
Nonetheless,
clearly
any
classically
defined
linear
differential
operator
is
weakly
continuous.
Theorem
3 •
5.
3
Let
B
1
and
B
2
be
two
smooth
vector
bundles
over
M,
let
L: c-cM, B
1
) -+ c-cM, B
2
)
be
a
weakly
continuous
linear
map.
The
necessary
and
sufficient
condition
for
L
to
be
a
linear
differential
operator
of
orderS
m
is
that:
for
any
s e c-cM,B
1
),
p
eM,
and
fa
smooth
function
of
M,
then
there
exist
a
function
g
in
x
from
R
to
Rdim
B.a.,
g(x)
(p)
= e·ixfCp>[L(eixfs)
(p))
such
that
in
each
coordinate
it
is
a
polynomial
of
degree
S
m.
For
the
proof
we
refer
the
reader
to
[Hormander
1964;
Kahn
1980,
p.198).
We
wish
to
define
the
symbol
and
ellipticity
of
a
linear
differential
operator
before
we
discuss
nonlinear
differential
operators.
Locally,
the
symbol
of
a
linear
partial
differential
operator
of
order
m
defined
over
Euclidean
space
R"
can
be
thought
of
by:
(a)
ignoring
the
terms
of
order
less
than
m,
and
(b)
in
each
term,
replacing
a;axi
by
ti
thus
obtaining
a
form
in
the
variables
ti
with
smooth
functions
as
coefficients.
For
higher
derivatives,
they
are
written
as
a
power
of
tp
i.e.,
replace
ak;ax/
by
tik.
If
the
symbol
is
definite,
i.e.,
all
the
coefficients
are
positive,
or
in
other
words,
the
symbol
vanishes
only
when
all
the
variables
are
set
equal
to
zero,
then
we
say
that
the
linear
differential
operator
is
elliptic.
For
instance,
the
3-dim
Laplacian
a
a•;ax•
+
a•;ay•
+
a•;az•
has
symbol.,.
+
tz
2
+
t3
2
and
clearly
it
is
elliptic.
The
linear
operator
L =
aa•;ax•
+
ba•;ay•
-
ca;ax
+
dajay
has
symbol
at
1
•
+
bt
2
•
and
it
is
elliptic.
But
the
wave
operator
a•
;ax•
+
a•
;ay•
+
a•
;az•
- c"
2
a•
;at•
has
symbol
.,.
+ t
2
•
+ t
3
• - c"
2
t
4
•
is
not
elliptic.
The
term
elliptic
comes
from
the
theory
of
quadratic
forms
as
it
relates
to
conic
131
sections.
Thus,
there
are
also
linear
differential
operators
of
the
types
parabolic
and
hyperbolic.
For
instance,
the
wave
equation
is
hyperbolic,
and
the
diffusion
equation
is
parabolic.
In
order
to
treat
the
symbol
in
a
global
setting,
one
cannot
just
consider
a
single
form,
we
need
the
following
machinery:
Given
a
smooth
m-dim
manifold
M
and
two
smooth
vector
bundles
over
M,
(BpM,7ri),
(i
=
1,
2).
Let
T*M
be
the
cotangent
bundle
of
M
with
projection
p.
Let
P:
~(M,B
1
)
~
~(M,B
2
)
be
a
linear
differential
operator
of
order
k.
Let
us
now
consider
the
induced
bundle
p"
1
(B
1
)
= {
(u,v)
e
T*MxB
1
1
7r
1
(v)
=
p(u)}.
Let
a
eM,
n e M
8
*.
Let
f
be
a
smooth
function
in
a
neighborhood
of
a
where
f(a)
= 0
and
df(a)
=
n.
For
e e
7r
1
"
1
(a),
let
s
be
a
smooth
section
of
(B
1
,
M,
7r
1
)
such
that
s(a)
=
e.
Now
set
aP(n,e)
=
P(fks)
(a).
Since
n e M
8
*
and
e e
7r
1
-
1
(a),
clearly
(n,e)
e
p"
1
(B
1
).
We
call
the
map
aP:
p"
1
(B
1
)
~
B
2
the
symbol
of
the
differential
operator
P.
For
each
a
eM
and
n e M
8
*,
aP
is
a
map
from
71"
1
-
1
(a)
to
71"
2
"
1
(a).
A
linear
differential
operator
P
is
elliptic
if
for
each
nonzero
n e M
8
*, a e
M,
the
map
aP
is
one-
to-one.
As a
simple
example,
consider
the
two-dimensional
Laplacian
in
the
plane
A=
a•;ax•
+
a•;ay•
in
a
trivial
one-dimensional
vector
bundle
over
R
2
•
Then
7r"
1
(B)
is
a
trivial
one-dimensional
vector
bundle
over
R
2
xR
2
= R
4
•
For
a
given
1-form
a
in
R
2
,
a =
adx
+
bdy,
choose
f(x,y)=
ax
+
by.
Let
the
section
be
s(x,y)
((x,y),m),
where
m
is
a
variable.
Then
(a•;ax•
+
a•;ay•)((ax
+
by)
2
m)
=
2a
2
m +
2b
2
m.
If
a
and
b
are
not
both
zero,
this
is
clearly
a
one-to-one
map
in
terms
of
the
variable
m,
thus
A
is
elliptic.
As
another
example,
let
M
be
a
smooth
n-dim
manifold
and
let
Ak(T*M)
be
the
bundle
of
k-forms.
We
have
a
linear
differential
operator
of
order
1,
d :
ok
(M)
~
ok+
1
(M)
where
ok(M)
=
~(M,Ak(T*M))
are
smooth
differential
k-forms.
When
i
=
O,
it
is
easy
to
show
that
the
symbol
ad
of
d
is
elliptic.
132
Let
f
be
a
smooth
function
of
M,
a e M
8
*
and
(df)
(a)
a.
Let
s e D
0
(M)
= ce(M)
and
s(a)
=
e.
Then
in
a
local
coordinate
chart,
ad(a,e)
=
d(fs)
(a)
=
~i=l
(afs(a);ax;)dX;
=
e·a
+
~i=l
f(a)(as(a);ax;)dX;·
Since
we
may
choose
any
s
such
that
s(a)
=
e,
we
choose
s(x)
e
identically,
i.e.,
as(a)jax;
=
0.
Thus
ad(a,e)
=
ea,
and
d : D
0
(M)
-+ 0
1
(M)
is
elliptic.
With
the
same
information
as
provided
earlier,
it
is
known
[Ch.1
of
Palais
1965]
that
the
vector
spaces
for
an
elliptic
differential
operator
P
of
positive
order
on
a
compact
manifold
Ker
p =
{f
€
ce(M,B,)
I
P(f)
0}
and
Coker
P = C"'(M,B
2
)/
{g
e c"'(M,B
2
)
I g =
Pf}
are
both
finite
dimensional.
The
analytic
index
of
P
is
an
integer
i
8
(P)
=dim
(Ker
P)
-dim
(Coker
P).
It
is
known
that
this
index
is
invariant
under
deformation
of
P,
and
thus
suggests
that
there
might
be
a
topological
description
of
i
8
•
This
has
led
to
the
theory
of
the
topological
index
and
the
theorem
of
Atiyah
and
Singer
which
asserts
that
these
two
indices
are
the
same.
For
the
definition
of
the
topological
index
see
[Ch.
1,
3,
4
of
Palais
1965],
analytic
index
see
[Ch.
1,
5
of
Palais
1965],
both
analytic
and
topological
indeces
on
unit
ball-bundle
or
unit
sphere
bundle
of
M
see
[Ch.15
of
Palais
1965],
and
the
index
theorem
see
(App.I
of
Palais
1965],
applications
see
[Ch.19
of
Palais
1965]
and
the
topological
index
of
elliptic
operators
see
(App.2
of
Palais
1965].
A
very
good
review
of
differential
operators
on
vector
bundles
is
Ch.
4
of
(Palais
1965].
A
good
source
of
reference
on
the
subject
of
Atiyah-Singer
index
theorem
is
the
book
edited
by
Palais
(1965].
For
those
readers
who
want
to
go
to
the
source,
the
series
of
papers
by
Atiyah
and
Singer
(1963,
1968]
are
recommended.
So
far
we
have
defined
linear
differential
operators,
the
symbol
of
a
linear
operator,
the
ellipticity
and
analytic
index
of
a
linear
differential
operator.
We
shall
133