Mukhi S., Mukunda N. Lectures on Advanced Mathematical Methods for Physicists
Подождите немного. Документ загружается.
82
Chapter
4.
Differentiable Manifolds II
points
in
M.
Then
the
map
z*
which we defined earlier sends forms
on
]1;1
to
forms
on
aD.
Thus,
by
won
the
RHS
of
the
above, we really
mean
z*w.)
Proof:
Choose a locally finite cover
(U
a
,
¢a)
of
M
and
a
partition
of
unity
eO.
subordinate
to
this
cover.
Then
On
the
RHS, we have a
sum
over forms,
each
of
which
has
support
in a single
U
a'
Thus,
it
is
enough
to
verify
the
theorem
for
such
forms, 'so we
take
w
to
have
support
in one
particular
U
a
.
Now we
can
prove Stokes'
theorem
in
each
of
two possible situations:
Case
1:
U
a
n
aD
=
¢
(the
open
set
does
not
intersect
the
boundary
of
D).
Then:
faD
W
=
0
Now, since
U
a
does
not
intersect
the
boundary
of
D,
it
either
lies entirely inside
or
entirely
outside
D.
Thus
U
a
C
M -
[)
or
U
a
C
D.
In
the
former case,
J
D
dw
=
0
and
we
are
done.
In
the
latter
case,
Since
w
has
compact
support,
we
can
take
the
integration
region
to
be
a
cube
in
IR
n
,
call
it
C,
of
side
2A,
such
that
al" =
0
on
the
border
of
C
(any
xl"
=
A).
Then
10
dw
=
10
(~avaV(X))dxl
..
·dx
n
=
L
10
aV(x
1
..
·xn)dx
1
..
·d:i;v
..
'dxnIXv=>,
=
0
v
Xv=-A
(4.71 )
Thus
we have shown
that
both
sides of Stokes'
theorem
vanish in
this
case,
hence
the
theorem
is
true.
Case
2:
U
a
n
aD
i=
¢.
Then,
since
aD
is defined
by
xn
=
0, we have
faD
w
=
(_1)n-l
J
an(x
1
, ...
,x
n
-
1
,
0)
dxl
...
dx
n
-
1
and
where now
the
cube
extends
from -
A
to
A
in
all directions
except
the
nth
one,
where
it
extends
from 0
to
A.
Thus,
for
any
fixed
j
i=
n
we have:
4.8.
The Laplacian on Forms
83
(there
is no
sum
over
v
in
the
above
equation),
while:
ic(Onan)dx1
..
·dx
n
=
(_l)n-l
J
an(x
1
,
...
,Xn-1,O)dx
1
..
·dx
n
-
1
Thus
we have shown
that
both
sides
of
Stokes'
theorem
are
equal
to
the
same
expression.
This
finally proves
the
theorem.
4.8
The
Laplacian
on
Forms
For a
p-form
W
=
W!"l
...
!"p
(x)
dX!"l
1\ .
..
I\dx!"p
,
we have
already
defined
the
Hodge
dual:
*W
=
V§
!"l
···.u
p
( ) d
!"p+l
1\ 1\
d
!"n
(n _
p)!
E
!"p+l"'!"n W!"l"'!"p X X
...
x
The
Hodge
dual
enables us
to
define
an
inner
product
on
forms
as
follows:
(a,
(3)
=
r
a
1\
*(3
JM
Since
a
and
(3
are
p-forms,
a
1\
*(3
is
an
n-form, hence as we have
just
seen,
it
makes sense
to
integrate
it
over
the
manifold
!v!
(we
assume
that
M
has
no
boundary.)
The
inner
product
thus
takes
two p-forms, for
any
p,
and
gives a
number.
In
components,
1\
*(3
-
In
!"lVl...
!"pV
p
(3
d
1
1\
...
1\
d
n
a -
ygg
9
a!"l"'!"p
Vl"'Vp X X .
so
the
inner
product
of
the
forms is
the
integral
over
the
inner
product
of
the
components,
with
all indices
contracted
via
the
metric,
and
with
a weight factor
V§
which ensures
coordinate-independence
of
the
result.
From
the
above
equation
it
is clear
that
the
inner
product
is symmetric:
(a,
(3)
=
((3,
a)
and
moreover,
the
inner
product
of
a form
with
itself is
greater
than
or
equal
to
zero:
(a,a)
~
0
Since
the
integral
of
a
strictly
positive
quantity
can
vanish only if
the
quantity
itself vanishes,
this
can
be
zero only if
the
the
square
of
the
form
components
is zero
at
each
point,
which
in
turn
means
the
form itself is identically zero.
Given
the
exterior
derivative,
the
inner
product
allows
us
to
define its
adjoint,
denoted
5.
This
is defined
by
(a,
d(3)
=
(5a,
(3).
Going
to
components,
one finds
84
Chapter
4.
Differentiable Manifolds II
where
the
sign
depends
on the
product
of
n
(the
dimension
of
the
manifold)
and
p
(the
dimension
of
the
form).
Thus,
the
adjoint
operator
15
can
be
thought
of
as
±
* d
*.
The
action
of
15
lowers
the
dimension
of
a form by one
unit,
the
opposite of
what
d
does.
If
W
is a
p-form,
then
*w
is
an
(n
- p)
form,
d*w
is
an
(n
- p
+
1) form
and
*d*w
is
an
(n - (n -
p
+
1))
=
(p -
1) form.
Example:
In
the
particular
case
of
three
space dimensions (n
=
3), we find
that
the
d-operator
on
I-forms is
the
familiar curl:
. . 1
..
d:
w
=
Wi
dx'
--+
dw
=
2(OiWj - OjWi)
dx'
1\
dx
J
We
can
now check
that
15
on
I-forms is
the
divergence:
. 1 .
15:
w=widx'
--+
-O;(y'gw')
v0
(4.72)
Exercise:
Derive
the
above result
and
show
that
the
RHS is a genuine scalar
under
coordinate
transformations.
Clearly
Jw
defines
the
generalization
of
the
divergence
Oi
wi
to
an
arbitrary
Riemannian
manifold
in
3 dimensions.
Just
as
d
2
=
0, we also have
15
2
=
0:
JJw
=
±(*d*)(*d*)w
=
±(*d
2
*)w
=0
(4.73)
A very
important
second-order differential
operator
in
familiar, fiat space
physics is
the
Laplacian
~
=
Oi
Oi.
The
generalization
to
a manifold is defined
using
the
d
and
15
operators.
Definition:
The
Laplacian
~
is defined
by
~
=
(d
+
15)2
=
dJ
+
Jd
The
equality
of
the
two expressions is
of
course due
to
d
2
=
15
2
=
o.
Clearly
the
Laplacian
(unlike
d
and
15)
maps
p-forms
back
to
p-forms:
Moreover
it
is self-adjoint in
the
norm
on
p-forms.
Thus,
for example,
it
makes
sense
to
talk
of
eigenvectors
and
eigenvalues
of
the
Laplacian
on
a manifold.
The
study
of
these
is called
harmonic
analysis.
In
the
special case
of
Euclidean
space
lR
n
and
in
Cartesian
coordinates,
the
Laplacian
on
any
p-form
gives:
~w
=
(OjOj Wi1".i
p
(x))
dX
i1
1\
...
1\
dx
ip
4.8.
The
Laplacian on Forms
85
Thus
in this case it is indeed
the
usual Laplacian
8/J
i
on
the
individual com-
ponents.
The
Laplacian
is
a positive
operator,
which means its
matrix
elements
between any
pair
of
forms
is
greater
than
or equal
to
zero:
(w,
b...w)
=
(w,
d5w
+
5dw)
=
(5w, 5w)
+
(dw,
dw)
::::
O.
It
also follows from
this
that
b...w
=
0 if
and
only if
5w
=
0 and
dw
=
0
(4.74)
At
this
point
we introduce some terminology
and
then
state
an
important
theorem.
Definition:
A form w is
said
to
be
closed if
dw
=
0,
and
exact if w
=
df
for
some
other
form
f.
Similarly, w is said
to
be
co-closed if
5w
=
0
and
co-exact
if w
=
5 f for some
other
form
f.
Finally, a form satisfying
b...w
=
0 is
said
to
be
harmonic. Clearly a form
is
harmonic if
and
only if it
is
both
closed
and
co-closed.
Hodge
decomposition
theorem:
If
M
is
a
compact
manifold
without
bound-
ary,
then
any
p-form
w
can
be
uniquely decomposed as
the
sum
of
exact, co-exact
and
harmonic forms:
w
=
da
+5{3+,
for some forms
a,
{3",
where
b...,
=
O.
Clearly if w
is
a p-form,
then
a,
{3"
are
(p
-
1),
(p
+
1)
and
p
forms respectively.
We omit
the
proof
of
this
theorem
here as
it
is
somewhat
involved. Never-
theless
the
result will
be
a useful tool in
studying
de
Rham
cohomology in
the
following chapter.
This page intentionally left blankThis page intentionally left blank
Chapter
5
Homology
and
Cohomology
5.1
Simplicial
Homology
We have seen in
Chapter
2
that
the
topological properties any topological space
can
be
understood
to
some
extent
via
homotopy,
the
study
of loops in
the
space.
An
alternative
approach
to
studying
topological properties, for a differentiable
manifold, arises
through
the
study
of objects called
simplices.
This
can
be
used
to
characterize topological properties of manifolds in
terms
of
simplicial
homology.
A closely related methodology,
though
from a completely different
starting
point, is
to
characterise topology
through
the
study
of differential forms.
This
goes by
the
name
of
de
Rham
cohomology.
The
two approaches are in a
certain
sense
dual
to
each other, as we will explain.
We
start
by developing
the
theory
of simplices
and
simplicial complexes.
Intuitively,
the
idea
is
to
define nice subsets of Euclidean space which look like
polyhedra.
These
can
be used
to
cover a manifold by a process called
triangula-
tion.
Topological properties of
the
manifold
can
then
be expressed in
terms
of
the
pieces which
triangulate
it.
To develop some
intuition
about
this
process, a simple example is provided
by
taking
the
2-sphere 3
2
and
drawing triangles all over it
to
completely cover
it.
Each
of
the
triangles has one edge in common
with
some
other
triangle
(Fig. 5.1)
The
triangles in
the
figure are triangles only in
the
sense
that
they
are
bounded
by
three
lines. These
are
not
"straight
lines" in
any
sense,
and
the
answers we
extract
will
not
depend
on
any local details of
the
lines or triangles.
The
important
thing, as we will see in
what
follows,
is
that
a generalised version
of
triangulation
provides a powerful tool for
the
formulation
and
solution of
problems regarding
the
topological properties of a manifold in
any
number
of
dimensions.
For
the
general case,
we
first define
an
object
called a simplex.
This
is
defined in Euclidean space lR
n
with
no reference
to
the
manifold of interest.
If
Xl,
X2,···
,
XM+l
are
distinct
points in lR
n
,
they
are said
to
be
linearly indepen-
88
Chapter
5.
Homology
and
Cohomology
Figure 5.1: A
triangulation
of 8
2
.
dent
if
the
M vectors
X2
-
Xl,
X3
-
Xl
,
"',
XM+l
-
Xl
are
linearly independent
vectors. A simplex will
be
obtained
by "filling in"
the
region in
ffin
defined by
a set of such points.
Definition:
An M
-simplex
aM
is
the
set of points
M+l
aM
= {
X
E
ffin
I
X
=
L
Ai
Xi,
i=l
M+l
L
Ai
=1,
i=l
where
Xl,
.
..
,
XM+l
are
linearly independent as defined above.
Here
are
some examples of simplices:
(i) A I-simplex in
ffi3
is
the
set
with
Al
+
A2
=
1.
Thus,
it
is
the
collection of points:
as
Al
varies from 0
to
1.
Clearly,
this
is
the
straight
line joining
Xl
to
X2.
(ii) A 2-simplex in
ffi3
is
the
set
with
Al
+
A2
+
A3
=
1.
Thus,
X
=
AlXl
+
A2X2
+
(1
-
Al
-
A2)X3
(5.1)
(5.2)
(5.3)
(5.4)
(5.5)
This
is
just
the
interior of
the
triangle,
including
the
boundary. Linear indepen-
dence of
the
points
Xl,
X2,
X3
ensures
that
they
are
not
collinear.
In
general,
the
points
{Xl,'"
,XM+d
are
called
the
vertices
of
the
M-
simplex
aM
,
which itself is denoted
(5.6)
Note
that
an
M-simplex is
not
just
the
set
of vertices
but
the
whole region
"enclosed"
by
them.
What
we have defined
are
more accurately described
as
5.1. . Simplicial Homology
89
X2
Figure
5.2:
The
shaded
region (including
the
boundary)
is
an
example of a
2-simplex.
"closed simplices".
It
is evident
that
each
M-simplex
is homeomorphic toJO, I]@
[0,1]
@
...
@
[0,1]
(M
times),
or
in
other
words,
to
a closed subset of lR .
The
set
of
numbers
Ai
which label points in
the
simplex are called
barycen-
tric coordinates.
To
understand
this
terminology, observe
that
if masses
Ai,
i
=
1"
..
,M
+
1
are
placed
at
the
points
Xi,
(i
=
1",
.
,M
+
1)
then
the
centre
of
mass is
located
at
X
=
L:~tl
Ai
Xi,
if
L:i
Ai
=
1.
For each M -simplex, we
can
define
the
faces
as
the
collection of
(M
- 1)-
simplices which lie
"opposite"
the
vertices. For example, for a triangle, we
can
associate
to
each vertex
the
line
that
lies opposite it.
Thus
we have:
M+l
Definition:
The
jthface
of
an
M-simplex
(TM
is
the
(M
-I)-simplex
L
AiXi
i=l,
i=/:-J
M+l
with
L
Ai
=
1.
This
amounts
to
considering
the
subset of
the
M -simplex
i=l,
i=/:-J
obtained
by
setting
Aj
=
O.
Since
an
M-simplex
has
M
+
1 vertices,
it
evidently
also has
M
+
1 faces.
Definition:
An
open simplex
is
the
interior
of
any closed simplex. We denote
this
by
((TM).
Given independent points
Xl,X2,'" ,XM+l ,
an
open
simplex is
M+l
(TM
= {
X E
IR
n
I
X
=
L
Aixi,
(5.7)
i=l
i=l
We have simply modified
the
definition of closed simplex by restricting
to
Ai
>
0
instead
of
Ai
~
O.
Clearly
an
open
simplex is
an
open
set in.
the
corresponding
Euclidean space.
Finally, we define a simplicial complex as a collection of simplices
with
specific properties.
Definition:
A
simplical complex
K
is
a finite collection of (closed) simplices in
some
IR
n
satisfying:
(i)
If
(TP
E
K,
then
all faces
of
(TP
belong
to
K.
(ii)
If
(TP,
(Tq
E
K,
then
either
((TP)
n
((Tq)
=
¢,
or
(TP
=
(Tq.
90
Chapter
5.
Homology
and
Cohomology
(Recall
that
by
((YP)
we
mean
the
open
simplex corresponding
to
the
closed
simplex
(Yp.)
The
dimension
of a simplicial complex
K
is defined
to
be
the
dimension of
(YP
E
K
for
the
largest
p.
Figure
5.3: A two-dimensional simplical complex which is
the
union
of
three
O-simplices,
three
I-simplices
and
one 2-simplex.
Basically
the
first
part
of
the
definition says
that
if
a
particular
simplex
belongs
to
a simplicial complex,
then
that
complex
must
necessarily contain all
the
lower dimensional simplices in
the
original one.
The
second
part
says
that
any
two distinct simplices in
the
complex
can
only
touch
along components
of
their
faces,
but
cannot
actually overlap each other.
An
example of a 2-dimensional simplicial complex
can
be
defined
by
its
collection
of
simplices as follows:
This
complex is
illustrated
in Fig. 5.3.
Some more possible two-dimensional simplicial complexes are
illustrated
in
Fig. 5.4, while Fig. 5.5 shows some objects
that
are
not
simplicial complexes.
For example, in
the
first two diagrams of Fig. 5.5 a O-simplex is missing. (In
these
figures, heavy
dots
indicate O-simplices.)
Figure
5.4: Examples of simplicial complexes (check
that
the
axioms hold!).
We still need a few more definitions before we
can
use
simplicial complexes
5.1. Simplicial Homology
91
--<
Figure
5.5: These
are
not
simplicial complexes (O-simplices
and
I-simplices are
missing in some of these diagrams).
to
study
the
topology of differentiable manifolds.
Definition:
The
union
of
all members of a simplicial complex
K
with
the
Euclidean subspace topology is called
the
polyhedron associated
with
K.
(The
polyhedron will also
be
denoted
K,
whenever
there
is no chance of confusion.)
Definition:
A smooth triangulation
of
a differentiable manifold M is a home-
omorphism
¢:
K
---+
M
for some polyhedron
K.
(¢
must
satisfy a technical
property
that
we
will
not
go into here.)
So far,
this
has been a
rather
abstract
discussion
with
no reference
to
manifolds.
The
key result
that
relates all this
to
the
topic of
our
interest is
that
every compact
Coo
manifold can
be
smoothly triangulated.
This
is
the
basis
of
the
study
of
manifolds
through
simplicial complexes. We will
not
be
able
to
provide a
proof
of
this
result here.
We will need
to
refine
the
definition of a simplex
to
take
into account
the
concept
of
orientation:
Definition:
An
oriented p-simplex
(p
2:
1)
is a p-simplex along
with
an
order-
ing for
its
vertices.
The
equivalence class
of
even
permutations
of
the
chosen
ordering defines a positively oriented simplex
+o-P,
while
odd
permutations
give
the
negatively oriented simplex, denoted
-o-p.
(Oriented simplices should
be
denoted by a new symbol, such as
(o-P)
,
but
we avoid
it
to
save
notation
and
because henceforth
we
will always work
with
oriented simplices).
For example, a 2-simplex
0-
2
=
[
Vo,
VI,
V
2
]
is associated
to
the
positively
oriented simplex
(5.9)
and
the
negatively oriented simplex
(5.10)
Associated
to
every oriented p-simplex, we would like
to
define a set
of
oriented
(p
- 1) simplices called
the
boundary of
o-p.
As a
set
of
simplices,
the
boundary
is
just
the
collection of faces of
o-p.
But
we
need
to
assign
an