Mukhi S., Mukunda N. Lectures on Advanced Mathematical Methods for Physicists
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72
Chapter
4.
Differentiable Manifolds
II
d
2
=
O.
But
what
about
D2,
the
square of
the
covariant exterior derivative
D?
This
is
easily calculated
by
acting twice
with
D
on an
arbitrary
O(n)
vector
Aa:
( 4.42)
Definition:
The
curvature 2-form
is
defined by:
( 4.43)
which in components
may
be
represented:
1
R
a
=
_R
a
dx'"
II
dx
v
b
2
bll,l/
1
R
a
c
d
=
2"
bcd
e
II
e
( 4.44)
The
torsion
and
curvature
2-forms are fundamental
to
the
study
of
Riemannian
manifolds.
Exercise:
Check
the
following properties of T
a
and
R'/;:
(DT)a
==
dT
a
+
wab
II Tb
=
R
a
b
II
e
b
(DR)a
b
==
dRab
+
wac
II
R
C
b
+
WCb
II
R
a
c
=
0
The
second of these relations is called
the
Bianchi identity.
( 4.45)
Under
O(n)
rotations,
the
torsion
and
curvature
transform
respectively as
T
a
--+
AabT
b
R'/;
--+
AacAbdRcd
=
(ARA-1)a
b
( 4.46)
Let
us now
turn
to
a new
type
of connection. We have seen
that
a spin
connection
is
needed in order
to
differentiate
O(n)
tensors covariantly. Suppose
instead
that
we
wish
to
covariantly differentiate a
vector field
(4.47)
As before,
the
ordinary
derivative produces something
that
does
not
transform
as a tensor:
8
dA
=
8
AIL
dxv
@ -
v
8x'"
So again, we
postulate
a covariant derivative:
( 4.48)
(4.49)
4.3. Connections,
Curvature
and
Torsion
73
such
that
DvAI"
transforms
as a
tensor
under
changes
of
coordinates. Requiring
linearity, we assume
(
4.50)
The
quantity
r~A
is called
the
affine connection.
In
order
for
DA
to
trans-
form covariantly
under
general
coordinate transformations
xl"
_
x'l"
(x),
the
affine connection
must
transform
as
ax'l"
ax{3
ax
v
ax'l" a
2
x'"
r'l"
(x') -
----
__
r'"
(x)
+
---:----::-----:-
VA
- ax'" aX,vaX!)'
{3v
ax'" ax,vax'A
(4.51)
Exercise:
Check
that
with
the
above
transformation
law,
the
covariant deriva-
tive
of
a vector field
transforms
covariantly
under
general
coordinate
transfor-
mations.
On
I-forms
B
=
Bj.tdxj.t,
the
covariant derivative is defined similarly,
but
with
an
important
change
of
sign
and
different index contractions:
DB
=
DvBl"dxv
®
dxj.t
DvBj.t
==
avBI" -
r~vBA
( 4.52)
This
is required so
that
the
contraction
of a covariant
and
contravariant
tensor
behaves as a scalar.
Note
that
DB
as we have defined
it
above is
not
an
antisymmetric
2-form,
but
just
a general rank-2 covariant tensor. We could,
of
course,
take
the
anti-
symmetric
part
of
it
to
get
a 2-form.
In
that
situation,
recalling
the
properties
of
exterior derivatives, one would
not
need
any
connection
at
all.
From
the
above
equation, we find
that
the
antisymmetrised
D
acting
on
a
I-form
contains as a
connection
the
object
r~v
-
r~j.t'
which is called
the
torsion
associated
to
the
affine connection
4
.
It
is easy
to
check
that
this
torsion
transforms
as a tensor,
and
can
be
chosen
to
vanish consistently
with
the
covariance
properties
of
the
derivative.
Returning
now
to
the
spin connection
w,
we have observed above
that
its
symmetric
part
transforms
as a tensor.
Thus
it
is
not
really required
in
order
to
fulfill
the
main
objective
of
a connection, which is
to
provide a rule for covariant
differentiation. Therefore one
(standard)
way
to
specify
W
is
to
require
that
it
be
antisymmetric.
If
in
addition
we require
the
torsion
(the
covariant derivative
of
w)
to
vanish,
it
can
easily
be
seen
that
w
is completely
determined
in
terms
of
the
frames
ea.
To see this,
let
us impose:
antisymmetry:
Wab
=
-Wba
no torsion:
T
a
==
(De)a
=
0
( 4.53)
Exercise:
Show
that
the
above conditions imply
the
relation:
W~b
=
~Eav
[(aj.te~
-
ave~)
-
e~Eba(aveca
- aaecv)] -
(a..-+
b)
(4.54)
4This
is
distinct
from
the
torsion
defined
above
in
terms
of
orthonormal
frames.
74
Chapter
4.
Differentiable Manifolds II
The
w
so defined is sometimes known
as
the
Levi-Civita spin connection.
Similarly,
an
affine connection
r
can
be
uniquely specified
by
two condi-
tions:
metricity:
D/-Lgv)..
==
0/-Lgv).. -
r~vga)..
-
r~)..gva
=
0
no
torsion:
r~)..
-
r~v
=
0
Metricity
is
the
same
as
covariant
constancy
of
the
metric.
(4.55)
The
affine connection is uniquely
determined
in
terms
of
the
metric
by
the
two conditions above:
/-L
_
1
/-La
( )
r
v)..
-"2
g
gav,)..
+
ga)..,v
-
gv)..,a
( 4.56)
This
is called
the
Levi-Civita affine connection,
or
Christoffel symbol.
Exercise:
Demonstrate
that
Eq.(
4.56) follows from
the
two conditions
in
Eq.(
4.55)
above.
The
spin
connection
and
affine connection
can
be
related
to
each
other
by
requiring covariant
constancy
of
the
frames:
( 4.57)
This
equation
clearly
determines
either
connection in
terms
of
the
other
one.
Exercise:
Solve
the
above
equation
for
the
spin
connection
in
terms
of
the
frames
and
the
affine connection. Next, use
Eq.(
4.56)
to
express
the
affine
connection
in
terms
of
the
metric
and
thence
in
terms
of frames using
g/-LV
=
e~e~.
At
the
end, you will have
an
expression for
the
spin
connection
in
terms
of
the
frames. Check
that
this
is identical
to
Eq.(4.54).
From
all
the
above,
it
is
evident
that
both
the
spin
connection
and
the
affine connection
are
essential ingredients
in
any
system
where we would like
to
differentiate
vector/tensor
fields, differential forms,
and
O(
n)
tensors
on
a
manifold.
While
there
is a
certain
degree of
arbitrariness
in
their
definition,
there
are
certain
"minimal" conditions which
can
be
imposed
to
render
them
unique.
In
applications
to
physics,
it
will
turn
out
that
these
conditions
are
naturally
satisfied in
the
theories
of
interest.
The
corresponding
connections
are
then
dependent
variables,
being
determined
by
the
frames
in
the
case
of
the
spin
connection
and
by
the
metric
in
the
case
of
the
affine connection
5
.
4.4
The
Volume
Form
Recall
that
the
group
SO(n)
differs from
the
orthogonal
group
O(n)
in
that
in
the
former,
the
rotation
matrices
are
required
to
have
unit
determinant
in
SSome
physical
applications
appear
to
require
an
affine
connection
that
has
nonzero
torsion
(i.e. is
not
symmetric
in
its
lower
indices).
However,
since
the
torsion
so
defined
is a
tensor,
it
can
always
be
treated
separately
from
the
connection
and
this
choice
usually
proves
more
convenient
in
practice.
4.4.
The Volume Form
75
addition
to
being
orthogonal.
Thus,
reflection
of
an
odd
number
of
space di-
mensions, which
constitutes
an
orthogonal
transformation
disconnected from
the
identity, is
not
included
in
SO(n)
though
it
is
part
of
O(n).
Now for
an
orientable
n-dimensional manifold we
should
ignore
those
frame
rotations
that
are
not
continuously
connected
to
the
identity,
as
such
rotations
would reverse
the
orientation.
Therefore
we
restrict
ourselves
to
SO(n)
rotations
of
the
frames
rather
than
O(n).
Indeed, frames of a given
handedness
can
be
chosen continuously everywhere
on
an
orient able manifold. .
Consider
the
n-form:
( 4.58)
This
is
invariant
under
SO(n)
rotations,
as
the
following exercise shows.
Exercise:
Prove
that
when
e
a
--+
Aabeb
with
Aa
b
E
O(n),
A
changes
to
(det
A)A.
Hence if
det
A
=
1, (equivalently, A
E
SO(n))
then
A
is invariant.
Definition:
The
n-form
A
defined above is called a
Riemannian
volume form.
We will see
that
the
volume form plays a crucial role in
integration
theory
on
manifolds.
We
can
rewrite
the
volume form
as
A
=
e
l
e
2
...
en
dX/.Ll
1\
d
X
/.L2
1\ 1\
dx/.Ln
/.Ll
/.L2
/.Ln
...
=
(det
e)
dx
l
1\
dx
2
1\
...
1\
dxn
=
.;g
dx
l
1\
dx
2
1\
...
1\
dxn
( 4.59)
where
g
==
det
g/.Lv'
Another
useful way
to
write
it
is
in
terms
of
the
totally
antisymmetric
E-tensor, defined as:
E
/.Ll
.
00
/.Ln
=
0 (if two indices coincide)
=
+
1 (if
(Pl'
..
Pn)
is
an
even
permutation)
=
-1
(if
(Pl'
..
Pn)
is
an
odd
permutation)
(4.60)
(An even (odd)
permutation
means
that
(Pl'"
Pn) is
obtained
from
(1···
n)
by
an
even (odd)
number
of pairwise interchanges.)
In
terms
of
this
tensor
,
the
volume form
may
be
written
\ - y'g
d
/.Ll
1\ 1\
d
/.Ln
/\
-
-,
E/.Ll'
oo/.Ln
X
..
. x
n.
One
important
role of
the
volume form is
that
it
allows us
to
define a
"duality"
operation
which
maps
p-forms
to
(n -
p)-forms.
This
is
carried
out
via
the
Hodge dual
operation
*,
defined by:
*(dX/.Ll
1\
...
1\
dx/.Lp)
=
(n
~p)!
E/.Lloo·/.LP/.Lp+l·oo/.Ln
dX/.Lp+l
1\
...
1\
dx/.Ln
76
Chapter
4. Differentiable Manifolds
II
Equivalently, if
A
is a p-form:
then
*
A
is
the
(n -
p)
form satisfying
where
A
is
the
volume form.
It
follows
that
*A
Jfj
1-'1·
.I-'p
d
I-'
+1
d
I-'
=
(n _
p)!al-'l
...
l-'
p
E
I-'p+1···l-'n
X
p
II···
II
x
n
Since
the
above discussion is
somewhat
abstract,
let us give a couple
of
concrete
examples.
In
three
dimensions
with
the
Euclidean
(identity) metric,
the
volume form is:
A
=
dx
1
II
dx
2
II
dx
3
The
Hodge
dual
of
a
I-form
is a 2-form via:
*dx
1
=
dx
2
II
dx
3
(4.61)
( 4.62)
The
familiar "cross
product"
in
vector analysis makes use
of
this
dual. Given
two 3-vectors
11
and
W,
one defines two I-forms:
(4.63)
The
wedge
product
of
these
vectors is
the
2-form:
(4.64)
Finally, we define
the
Hodge
dual
of
this
2-form, which is a I-form:
_ ij k
*(VIIW)-E
kViWjdx
(4.65)
The
components
of
this
I-form
are
( 4.66)
which
are
precisely
the
components
of
the
vector
11
x
W,
familiar as
the
"Cross
product".
It
is unique
to
3 dimensions
that
the
wedge
product
of
two vectors
can
again, using
the
Hodge
dual,
be
expressed as a vector.
4.5
Isometry
In
preceding
chapters,
we have defined
the
criteria
that
two spaces
be
identical
as
topological spaces (homeomorphism)
and
as
differentiable manifolds (diffeo-
morphism).
Now we identify
the
property
which makes two spaces equivalent
as
Riemannian
manifolds.
Definition:
A
map
¢:
X
--+
Y,
where X
and
Yare
two
Riemannian
manifolds,
is
an
isometry
if:
4.6.
Integration
of
Differential
Forms
77
(i)
¢
is a homeomorphism.
(ii)
¢
is
Coo
in
both
directions.
(iii)
The
differential
map
¢* :
Tp(X)
---7
Tp(Y)
preserves
the
metric.
The
first
two
requirements
amount
to
saying
that
¢
is a diffeomorphism
of
the
two manifolds, while
the
last
requirement
says
that
if
A
and
B
are
any
two
vectors in
Tp(X),
then
(¢*A,
¢*B)c
=
(A,
B)c
If
two
Riemannian
manifolds
admit
an
isometry
between
them
then
they
are
said
to
be
isometric,
and
are
identical
in
topology, differentiable
structure
and
metric. We consider
them
to
be
equivalent.
4.6
Integration
of
Differential
Forms
We have seen
that
on
an
n-dimensional
manifold,
an
n-form
has
a single inde-
pendent
component
by
virtue
of
total
antisymmetry
in
n
indices.
In
fact,
the
dual
of
an
n-form
is a O-form
or
function. We
can
represent
any
n-form
as:
W
=
w/Ll
...
/Ln
(x)
dX/Ll
/\
...
/\
dx/L
n
=
a(x)
dx
1
/\
...
/\
dxn
(4.67)
where
( )
_
/Ll···/Ln
( )
a x -
f
w/Ll
...
/Ln
X
We would now like
to
define
integration
on
a
Riemannian
manifold.
For
this, we
assume
familiarity
with
the
integral
of
ordinary
functions
on
Euclidean
space, which is
the
same
as
the
integral
of
a function
of
many
variables over
some
range
of
its
arguments.
The
more
general
integration
we will now define
has
to
be
independent
of
the
coordinate
charts
we use
to
cover
the
manifold,
and
invariant
under
isometries
of
the
metric. As we will see,
the
natural
object
possessing
these
properties
is
the
integral
of
an
n-form
over
an
n-dimensional
manifold.
We will require
the
manifold
to
be
orientable.
This
requirement
can
be
guessed
at
from
the
fact
that
an
integral
of
an
ordinary
function changes sign
when
we reverse
the
integration
limits,
or
equivalently
when
we reverse
the
integration
measure.
The
generalisation
of
this
property
will
be
that
our
integral
changes sign
under
reversal
of
the
orientation
of
the
manifold.
To define
the
integral
of
an
n-form
on
an
orient
able
n-dimensional
manifold,
we
recast
the
problem
in
terms
of
the
usual
integration
of
the
function
a(x)
that
is Hodge
dual
to
the
n-form
over
suitably
subsets
of
lRn.
The
latter
integral
is
just
the
usual
Riemannian
one.
Combining
the
subsets
will
be
done
in
a way
that
fulfils
the
requirements
spelt
out
above.
Thus,
we
start
off
with
a
particular
chart
(Me>:)
¢o.)
on
M
and
define
the
integral
of
our
n-form
over
this
chart
as:
r
W
=
r
a(x)dx
1
...
dx
n
1Ma
1q,a(M
a
)CRn
78
Chapter
4.
Differentiable Manifolds II
The
right
hand
side is, as promised,
the
ordinary
integral
of
a function
of
several
variables over some
range
of
these
variables.
Note
that
a(x)
is
not
a scalar function
on
the
manifold, since
w
is coordinate-
independent
while:
It
follows
that
under
coordinate
transformations,
a(x)
--7
a'(y)
=
II
~::
II
a(x)
or
in
words,
a(x)
transforms
by a multiplicative
Jacobian
factor. Such
an
object
is sometimes called a
scalar density.
This
transformation
of
a(x)
precisely cancels
the
variation
under
coordinate
transformations
of
the
Cartesian
integration
measure
dx
1
...
dxn.
Thus
the
integrand
a(x)
dx
1
...
dxn
is
coordinate
independent.
This
explains why we
started
with
an
n-form, whose Hodge
dual
is a scalar density,
instead
of
simply
trying
to
integrate
a scalar function.
Suppose now we
take
another
chart
(Ma,
'l/Ja),
namely
the
same
open
set
lYla
but
a different homeomorphism
'l/Jo.
in place
of
the
original
cPo..
We
then
see
that:
r
a'(y)
dyl
...
dyn
=
r
a(x)
dx
1
...
dxn
J"'o.(Mo.)CRn
J"'o.(Mo.)CRn
Thus,
J
M
W
on
a
given
open
set
of
M
is invariantly defined.
It
only remains
to
put
the
contributions
from different
open
sets
together
to
get
an
integral over all
of
M.
In
order
not
to
encounter
divergences in
the
process, we confine ourselves
to
n-forms
with
compact support.
The support
of
w
is defined as:
Supp(w)
=
closure
of
{x
E
M
I
w(x)
=1=
O}
If
this
set
is
compact,
as defined in
Chapter
1,
then
w
is said
to
have
compact
support.
Alternatively,
instead
of
worrying
about
forms
with
compact
support
we
may
simply confine
our
attention
to
integration
over
compact
manifolds. Since
every closed
subset
of
a
compact
manifold is
compact
(see
Chapter
1), we
are
then
guaranteed
compact
support.
Finally we
turn
to
the
actual
process
of
patching up
the
integral
of
w
over
various
open
sets
of
M.
This
is achieved by defining a special collection
of
functions
on
M
called a
partition
of
unity.
Roughly,
we
will decompose
the
constant
function 1
on the
manifold into a
sum
of
functions each having
support
only within a given
open
set
in
the
cover.
This
construction
will enable us
to
patch
together
the
integrals
of
a form
on
the
various
charts
Mo.,
which we have
defined above,
to
give
an
integral over all
of
M.
4.6.
Integration
of
Differential
Forms
79
To implement this, first
we
need a few definitions.
Definition:
An
open
covering
{U
oj
of
M
is said
to
be
locally finite
if each
point
p
E
M
is
contained
in a finite
number
of
U
a
.
A space is
paracompact
if every
open
cover
admit
a locally finite refinement. (A refinement is defined as follows:
If
{U
a
}
is
an
open
cover,
then
another
open
cover
{V,e}
is a
refinement
of
{U
a
}
if every
set
Vj3
is contained in some
set
U
a
.)
Evidently,
paracompactness
is a
weaker requirement
than
compactness.
Now
take
a
paracompact
manifold
M
(incidentally, every me
tric
space is
paracompact)
and
a locally finite
atlas
{U
a
,
<Pa}.
Pick a family
of
differen-
tiable functions
ea(x)
on
M.
Each
e
a
is chosen
to
be
nonzero only
within
the
corresponding
open
set
U
a
.
The
precise requirements are:
(i)
0::;
ea(x)
::;
1,
for each a
and
all
x
E
U
a
.
(ii)
e
a
are
Coo
functions.
(iii)
ea(x)
=
0 if
x
'I-
U
a
·
(iv)
La
ea(x)
=
1.
Note
that
the
sum
in (iv) contains finitely
many
terms
for each
x,
precisely
because
the
cover is locally finite.
The
collection
of
functions
e
a
is called a
partition
of
unity subordinate
to
the open cover {U
a
,
<Pa}.
Given such a
partition
of
unity
, we
can
finally define
the
integral
of
an
n-form over
the
entire manifold
M.
Definition:
The
integral
of
an
n-form
w
over a manifold
Mis
:
In
order
for
this
definition
to
have
an
intrinsic meaning,
it
must
be
inde-
pendent
of
the
choice
of
the
partition
of
unity
and
the
atlas. We now prove
that
this is
the
case.
In
the
process,
the
meaning
of
the
definition should become
clearer.
Given two
partitions
of
unity,
e
a
subordinate
to
(U
a
,
<Pa)
and
e~"
subordi-
nate
to
(U~"
<p~,)
we have
(4.68)
where in
the
second line we were able
to
extend
the
range
of
integration
to
all
of
IR
n
since
e
a
has
support
only in
<Pa(U
a
).
80
Chapter
4.
Differentiable Manifolds
II
Now we multiply by I
=
Lal
e~,
to
get
=
L
1
e~,(x')a'(x')dx!l··
·dx
ln
0:'
Rn
(4.69)
In
the
third
step
we used
the
coordinate
independence
of
a(x) dx
1
...
dxn
while
in
the
last
step, we used
the
fact
that
e~,
has
support
in
U~,.
This
proves
the
desired result.
4.7
Stokes'
Theorem
For a
suitable
region in lR
n
,
Stokes'
theorem
is a familiar result which relates
the
integral
of
a divergence over
the
region,
to
the
integral
of
the
corresponding
function over
the
boundary
of
the
region.
The
generalization
to
manifolds, which
we now develop, is
an
important
theorem
and
will
reappear
in
the
context
of
de
Rham
cohomology
which we will
study
later
on.
Let
w
be
an
(n -
I)-form
on an
n-dimensional manifold:
W
=
wJ.Ll
...
J.Ln-l
(x)
dXJ.Ll
1\
...
1\
dxJ.Ln
-l
The
Hodge
dual
of
W
is a
I-form
with
components
which we will
denote
aV(x).
Then
we
can
write:
n
W
=
L(
-It-1av(x)
dx
1
1\
...
1\
dx
v
1\
...
dxn
v=l
(dxV
means
dx
v
is deleted).
Now
the
exterior
derivative
of
w
is
an
n-form, so its
dual
will
be
a O-form.
4.7.
Stokes' Theorem
We have:
n
dw
=
L aI"Wl"l"'l"n_l (x) dxl"
/I.
dXl"l
/I.
...
/I.
dxl"n
1"=1
n n
=
(LavaV(X))dX1
/I.
...
/I.
dxn
v
81
(4.70)
(The
last
equation
looks non-covariant: in fact,
both
ava
v
and
dx
1
/I.
...
/I.
dxn
are
not
coordinate independent,
but
the
product
is, since it is
dw).
Thus
we see
that
the
exterior derivative of
an
(n-l)-form
can
be expressed
in
terms
of
the
divergence of
the
components of
the
dual
I-form. To
set
up
Stokes'
theorem
we
next
have
to
define suitable regions in a manifold.
Definition:
Let
M be a differentiable manifold. A subset D of M
is
called a
regular
domain
(or simply
domain)
if for each
p
E
fJ
(recall
that
fJ
denotes
the
closure of
D),
either one of
the
following holds:
(i)
There
exists a local
chart
(U,
¢)
of
M
with
p
E
U,
such
that
¢(fJ
n
U)
is
open
in
lRn,
or
(ii) No such
chart
exists,
but
there
exists a
chart
(U,
¢)
with
p
E
U
such
that
¢(
fJ
n
u)
is
an
open
set of
the
half-space
Points
p
E
D
satisfying (i) are called
interior
points
of
D,
while points
satisfying (ii)
are
called
boundary points
of
D.
The
reason for
this
terminology
should be evident from
the
definition.
The
set of
boundary
points of
D
is denoted
aD
and
defined as
aD
= {
p
E
fJ
I
xn
=
0 }
It
is
a
theorem
(whose
proof
we
skip here)
that
aD
is
an
(n -
1) dimensional
orientable
sub
manifold of
D.
We now have
the
necessary ingredients
to
state,
and
prove, Stokes' Theo-
rem.
Theorem
(Stokes):
Let
M
be
an
oriented n-dimensional manifold
and
let
D
be
a regular domain in
M.
Let
W
be
an
(n -
I)-form
on
M
with
compact
support.
Then
l
dw
=
laD
W
(Strictly speaking,
W
is
a form on M
and
not
on
aD.
But
we have
an
injective
map,
the
'inclusion
t:
aD
---+
M
which maps points in
aD
to
the
corresponding