Eschrig H. Topology and Geometry for Physics
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Thus, locally an analytic Riemannian metric on a two-dimensional real mani-
fold can always be brought into the form (9.98). Coordinates x; y in this form are
called isothermal coordinates. If two metrics g
1
and g
2
are related as g
2
¼ fg
1
where f is a positive function, then it is readily seen from (9.4, 9.5) that all angles
have the same values in both metrics. The Riemannian geometries with such two
metrics are called conformal, and mappings F : M
1
! M
2
between geometries M
1
and M
2
which preserve all angles are called conformal mappings. Hence, an
analytic Riemannian metric on a two-dimensional manifold is always locally
conformal to the Euclidean metric. If there are coordinates w ¼ u þiv in which
ds
2
may be written as ds
2
¼j/ðwÞj
2
dwd
w, then either dw is proportional to dz
(orientation preserving) or to dz (orientation reversing). In the first case w is an
analytic function of z.IfFðzÞ is an analytic complex function with dF=dz ¼ f ðzÞ,
then locally ds
2
¼ dFd
F and Re F and Im F are Cartesian local coordinates in the
above considered manifold M.
The metric (9.98) defines a one-dimensional complex Riemannian manifold
which is also called a Riemannian surface. Note that ðx
1
þ ix
2
Þ¼fdzmeans
x
1
x
¼ x
2
y
¼ Re f ; x
1
y
¼x
2
x
¼ Im f . It is seen that every two-dimensional orient-
able real manifold allows for a complex manifold structure with a locally
Euclidean metric that makes it into a Riemannian surface.
Let M be a complex manifold of dimension m and consider the tangent space
T
z
ðMÞ on M at point z ¼ðz
1
; ...; z
m
Þ. The linear mapping J : T
z
ðMÞ!T
z
ðMÞ :
JX ¼ iX or JX ¼iX has the obvious property J
2
¼ J J ¼Id
T
z
ðMÞ
. Conversely,
if V is a vector space, then a linear transformation J : V ! V; J
2
¼Id
V
is called
a complex structure on V. Treat V as a real vector space, then two vectors X and
JX cannot be proportional to each other, JX ¼ kX with real k, because then
J
2
X ¼ k
2
X; k
2
0, against the assumption. Hence, X and JX span a two-dimen-
sional subspace E
X
of V which is clearly invariant under J; JE
X
¼ E
X
, and every
invariant subspace of J in V is two-dimensional. A complex structure can only
exist on V, if as a real vector space it is even-dimensional. If V
is the dual space
of V, then a complex structure J on V induces a complex structure (also called J)
on V
by the definition
hx; JXi¼hJx; Xi for all x 2 V
; X 2 V: ð9:99Þ
From this definition, J
2
¼Id
V
readily follows.
An example from physics is the time reversal operator
b
T in quantum
mechanics. As an operator in the space of spinor quantum states W it has the
property
b
T
2
¼Id and hence is a complex structure. If a Hamiltonian
b
H com-
mutes with
b
T, then
b
HW ¼ WE implies
b
Hð
b
TWÞ¼
b
T
b
HW ¼ð
b
TWÞE, and all
eigenvalues E of
b
H are twofold degenerate. This is called Kramers degeneracy.
Much more general implications of a complex structure on quantum physics are
discussed in [6].
9.7 Complex, Hermitian and Kählerian Manifolds 337
Treat further on V as a real even-dimensional vector space provided with a
complex structure J. Consider the set of complex valued linear functions on V as
the complexification V
C of the dual space V
. Since V
is a real vector space
of dimension m, V
C is a complex vector space of complex dimension m. Any
element X 2 V
C can be written as X ¼ x
1
þ ix
2
; x
j
2 V
. Extend J from V
to V
C by complex linearity. The linear operator J has eigenvectors as elements
of V
C with eigenvalues i, which are called elements of type ð1; 0Þ (upper
sign) and ð0; 1Þ (lower sign), respectively. Denote the subspace of V
C of all
elements of type ð1; 0Þ by V
C
and the subspace of all elements of type ð0; 1Þ by
V
C
.
Let X be any vector in V
C and let X
1
¼ð1 iJÞX=2; X
2
¼ð1 þ iJÞX=2.
Then, X ¼ X
1
þ X
2
and JX
1
¼ðJ þ iÞX=2 ¼ iX
1
; JX
2
¼ðJ iÞX=2 ¼iX
2
.
Hence, X
1
2 V
C
; X
2
2
V
C
and V
C ¼ V
C
V
C
. Moreover, J maps V
C
onto
V
C
under complex conjugation. Indeed, let again X ¼ x
1
þ ix
2
and let it be in V
C
.
Then, JX ¼ iðx
1
þ ix
2
Þ¼x
2
þ ix
1
. Hence, Jx
1
¼x
2
; Jx
2
¼ x
1
, and
J
X ¼ Jðx
1
ix
2
Þ¼x
2
ix
1
¼i
X, which means
X 2
V
C
for X 2 V
C
. Both
spaces V
C
and
V
C
are complex vector spaces of dimension m=2 each. If fX
j
jj ¼
1; ...; m=2g is any base in V
C
, then fX
j
;
X
j
jj ¼ 1; ...; m=2g is a base in V
C.
Let
X
j
¼ x
j
þ ix
m=2þj
2 V
C
;
X
j
¼ x
j
ix
m=2þj
2
V
C
; j ¼ 1; ...; m=2 ð9:100Þ
with
x
k
2 V
; k ¼ 1; ...; m. Since fx
j
¼ðX
j
þ
X
j
Þ=2; x
m=2þj
¼ðX
j
X
j
Þ=ð2iÞjj ¼
1; ...; m=2g is just another base in V
C with real base vectors x
j
; j ¼ 1; ...; m,
it is also a base in the real space V
. Because JX
j
¼ iX
j
; J
X
j
¼i
X
j
,
Jx
j
¼x
m=2þj
; Jx
m=2þj
¼ x
j
; j ¼ 1; ...; m=2: ð9:101Þ
The base fx
j
jj ¼ 1; ...; mg in V
is called a J-adapted base. For the dual base
fX
k
jk ¼ 1; ...; mg in V, hx
j
; X
k
i¼d
j
k
, this immediately implies
JX
k
¼ X
m=2þk
; JX
m=2þk
¼X
k
; k ¼ 1; ...; m=2: ð9:102Þ
It is likewise a J-adapted base. With these notations one has x
j
^ Jx
j
¼
x
j
^ x
m=2þj
¼ði=2ÞX
j
^
X
j
, whence the volume element is (with the obvious
notation ^
m
j¼1
x
j
¼ x
1
^^x
m
)
s ¼
^
m=2
j¼1
ðx
j
^ x
m=2þj
Þ¼
i
2
m=2
^
m=2
j¼1
ðX
j
^
X
j
Þ: ð9:103Þ
Any linear base transformation P
j
¼
P
m=2
k¼1
X
k
A
j
k
with a complex
ððm=2Þðm=2ÞÞ-matrix A yields ^
j
ðP
j
^
P
j
Þ¼jdet Aj
2
^
j
ðX
j
^
X
j
Þ and
hence it corresponds to a linear base transformation in V which preserves the
orientation of V.
338 9 Riemannian Geometry
Summarizing: If J is a complex structure on a real vector space V, then V is of
even dimension m, and there exists a base fX
k
; JX
k
jk ¼ 1; ...; m= 2g in V, and any
two such bases yield the same orientation of V: Conversely, if the dimension m of
V is even and there exists a direct sum decomposition V
C ¼ V
C
V
C
of the
complexified dual of V such that complex conjugation maps bijectively V
C
onto
V
C
; then there is a complex structure on V for which the elements of V
C
are of type
ð1; 0Þ and the elements of
V
C
are of type ð0; 1Þ:
For the last statement one simply puts JX ¼ iX in V
C
and JX ¼iX in
V
C
.
A ðp þ qÞ-linear alternating complex function on V which may be written as
X
j
1
\\j
p
;
j
1
\\
j
q
C
j
1
...j
p
;
j
1
...
j
q
X
j
1
^^X
j
p
^
X
j
1
^^
X
j
q
ð9:104Þ
is called an exterior ðp; qÞ-form. Obviously, if X is an exterior ðp; qÞ-form, then
X
is an exterior ðq; pÞ-from, and if P is an exterior ðr; sÞ-form, then X ^ P is an
exterior ðp þ r; q þsÞ-from.
Like the J-adapted base fx
j
jj ¼ 1; ...; mg could be used as a base in both the
real space V
and the complex space V
C, the J adapted base fX
k
jk ¼
1; ...; mg in V can also be used in the complex m-dimensional space V C. Then,
introduce in the latter space the alternative base
Z
k
¼
1
2
X
k
iX
m=2þk
;
Z
k
¼
1
2
X
k
þ iX
m=2þk
; k ¼ 1; ...; m=2 ð9:105Þ
and find hX
j
; Z
k
i¼hðx
j
þ ix
m=2þj
Þ; ðX
k
iX
m=2þk
Þi=2 ¼ d
j
k
¼h
X
j
;
Z
k
i; j; k ¼
1; ...m=2 since x
j
and X
k
were dual to each other for j; k ¼ 1; ...; m. Likewise one
finds hX
j
;
Z
k
i¼0 ¼h
X
j
; Z
k
i. The bases fX
j
;
X
j
g and fZ
k
;
Z
k
g are dual to each
other in the complex spaces V
C and V C.
Let V again be a real vector space with a complex structure J, and let H :
V V ! C be a complex function of two vector variables with the properties
(physics convention, mathematics convention would interchange the first with the
second argument, compare p. 19)
1: HðX; k
1
X
1
þ k
2
X
2
Þ¼k
1
HðX; X
1
Þþk
2
HðX; X
2
Þ; X; X
j
2 V; k
j
2 R;
2: HðY; XÞ¼
HðX; YÞ ; X; Y 2 V;
3: HðX; JYÞ¼iHðX; YÞ:
Then, H is called a Hermitian structure. The first two conditions mean that Re H
is a symmetric bilinear function, and Im H is an alternating bilinear function while
it follows from the second and third properties that
Re HðX; JYÞ¼Im HðX; YÞ; Im HðX; JYÞ¼Re HðX; YÞ;
HðJX; JYÞ¼Hð X; YÞ :
ð9:106Þ
9.7 Complex, Hermitian and Kählerian Manifolds 339
If besides the complex structure J a symmetric bilinear function FðX; YÞ on V is
given (or an alternating bilinear function GðX; YÞ), then it defines with the rela-
tions (9.106) a Hermitian structure Re H ¼ F (Im H ¼ G). The Hermitian struc-
ture H is called positive definite, if Re H ¼ F is a positive definite function. It is
easily seen that a positive definite Hermitian structure defines a (real) inner
product in V,
ðXjYÞ¼Re HðX; YÞ¼
1
2
HðX; YÞþHðY; XÞðÞ; ð9:107Þ
that is invariant under the mapping J, ðJXjJYÞ¼ðXjYÞ.
Expanded in the J-adapted base (9.102), general vectors X; Y 2 V are
X ¼
X
m=2
k¼1
n
k
X
k
þ n
m=2þk
JX
k
; Y ¼
X
m=2
k¼1
g
k
X
k
þ g
m=2þk
JX
k
; n
k
; g
k
2 R
and hence,
HðX; YÞ¼
X
m=2
j;k¼1
n
j
in
m=2þj
g
k
þ ig
m=2þk
HðX
j
; X
k
Þ:
Now, n
j
¼hx
j
; Xi, and hence, with (9.100), n
j
in
m=2þj
¼h
X
j
; Xi;
g
k
þ ig
m=2þk
¼hX
k
; Yi. Therefore,
HðX; YÞ¼
X
m=2
j;k¼1
H
jk
h
X
j
; XihX
k
; Yi; H
jk
¼ HðX
j
; X
k
Þ¼
H
k
j
; ð9:108Þ
that is, the Hermitian structure is given by a Hermitian 2-form
H ¼
X
m=2
j;k¼1
H
jk
X
j
X
k
ð9:109Þ
expressed in a J-adapted base in V
C:
The alternating 2-form (exterior 2-from) G with hG; X ^ Yi¼Im HðX; YÞ¼
HðX; YÞ
HðX; YÞ
=ð2iÞ¼
P
m=2
j;k¼1
H
jk
h
X
j
; XihX
k
; Yih
X
j
; YihX
k
; Xi
=ð2iÞ¼
ði=2Þ
P
m=2
j;k¼1
H
jk
X
j
^X
k
; X ^Y
DE
, that is,
iG ¼
1
2
X
m=2
j;k¼1
H
jk
X
j
^ X
k
; ð9:110Þ
is called the Kählerian form of the Hermitian structure H. Hence, H ¼ F þ iG,
where F ¼ Re H is a symmetric 2-form.
If V is a complex vector space of dimension m=2, then a Hermitian structure
on the complex vector space is a function H : V V ! C with the properties
340 9 Riemannian Geometry
1: HðZ; k
1
Z
1
þ k
2
Z
2
Þ¼k
1
HðZ; Z
1
Þþk
2
HðZ; Z
2
Þ; Z; Z
j
2 V; k
j
2 C;
2: HðZ
2
; Z
1
Þ¼HðZ
1
; Z
2
Þ; Z
j
2 V:
It is said to be positive definite, if HðZ; ZÞ[ 0. Thus, if V is an m-dimensional
real vector space with a complex structure J, then by defining iX ¼ JX for all
X 2 V it extends by complex linearity to a complex vector space of dimension m;
if it has a Hermitian structure H, then this is both a Hermitian structure of the real
and complex vector spaces V.
A real manifold M of dimension m with a given smooth real tensor field J of
type ð1; 1Þ such that at every point x 2 M the tensor J
x
provides the tangent space
T
x
ðMÞ with a complex structure is called an almost complex manifold and J is
called an almost complex structure on M. Clearly, M must be orientable and of
even dimension. However, it can be shown that not every orientable manifold of
even dimension can have an almost complex structure. Suppose M is an almost
complex manifold and x
j
are local coordinates in M. Then,
J
x
o
ox
j
¼
X
m
k¼1
J
k
j
ðxÞ
o
ox
k
;
X
m
l¼1
J
l
j
ðxÞJ
k
l
ðxÞ¼d
k
j
; ð9:111Þ
with smooth functions J
k
j
ðxÞ.
As will be seen now, every complex manifold is an almost complex manifold,
the opposite, however, is in general only true for m ¼ 2. Let M be an m=2-
dimensional complex manifold. It can be viewed as an m-dimensional real man-
ifold. Consider a coordinate neighborhood with local complex coordinates
ðz
1
; ...; z
m=2
Þ; z
j
¼ x
j
þ iy
j
; x
j
; y
j
2 R. Define the tensor field J by
J
z
o
ox
j
¼
o
oy
j
; J
z
o
oy
j
¼
o
ox
j
: ð9:112Þ
Obviously, J
2
z
¼Id
T
z
ðMÞ
. Of course, the definition of J must have a coordinate
independent meaning. Let z
j
¼ z
j
ðw
k
Þ be m=2 analytic complex functions in a
neighborhood of the point z 2 M, so that w
k
¼ u
k
þ iv
k
; u
k
; v
k
2 R are alternative
local coordinates there. Due to the Cauchy–Riemann conditions (9.97) for the
functions z
j
, one gets from (9.112)
J
z
o
ou
k
¼ J
z
X
j
ox
j
ou
k
o
ox
j
þ
X
j
oy
j
ou
k
o
oy
j
!
¼
o
ov
k
;
J
z
o
ov
k
¼ J
z
X
j
ox
j
ov
k
o
ox
j
þ
X
j
oy
j
ov
k
o
oy
j
!
¼
o
ou
k
:
Thus, J is correctly defined. This almost complex structure defined by (9.112)
is called the canonical almost complex structure of the complex manifold M.
From (9.99) it follows that
9.7 Complex, Hermitian and Kählerian Manifolds 341
J
z
ðdx
j
Þ¼dy
j
; J
z
ðdy
j
Þ¼dx
j
; ð9:113Þ
and hence dz
j
¼ dx
j
þ idy
j
is a complex differential ð1; 0Þ -form, J
z
ðdz
j
Þ¼idz
j
,
and dz
j
¼ dx
j
idy
j
is a complex differential ð0; 1Þ-form, J
z
ðdz
j
Þ¼idz
j
, both
as elements of the complex vector space T
z
ðMÞC ¼ T
z
ðMÞ
C
T
z
ðMÞ
C
.
Accordingly, in the complexified tangent space T
z
ðMÞC,
o
oz
j
¼
1
2
o
ox
j
i
o
oy
j
;
o
oz
j
¼
1
2
o
ox
j
þ i
o
oy
j
; j ¼ 1; ...; m=2; ð 9:114Þ
are tangent vectors of type ð1; 0Þ; J
z
ðo=oz
j
Þ¼iðo=oz
j
Þ; and ð0; 1Þ; J
z
ðo=oz
j
Þ¼
iðo=oz
j
Þ; respectively, which together span the complex tangent space
T
z
ðMÞC ¼ T
z
ðMÞ
C
T
z
ðMÞ
C
. The real cotangent space of M taken as an m-
dimensional real manifold is spanned by the combinations dx
j
¼ðdz
j
þ
dz
j
Þ=2; dy
j
¼ðdz
j
dz
j
Þ=ð2iÞ and likewise for the real tangent space. As is seen,
any complex manifold is indeed an almost complex manifold.
Conversely, let M be an almost complex manifold of (real) dimension m with
an almost complex structure given by the smooth tensor field J of type ð1; 1Þ.
Consider this almost complex structure locally given by m=2 linearly independent
differential 1-forms X
j
; j ¼ 1; ...; m=2 of type ð1; 0Þ and m=2 corresponding
differential 1-forms
X of type ð0; 1Þ. Let dx
j
¼ðX
j
þ
X
j
Þ=2 and
dy
j
¼ðX
j
X
j
Þ=ð2iÞ. If these differentials integrate locally to real analytic
coordinates (that is, to coordinates belonging to an analytic complete atlas with
only analytic transition functions of M), then M is a complex manifold. By virtue
of Frobenius’ theorem this is the case, iff
dX
j
0 mod X
k
; 1 k m=2
; ð9:115Þ
because this also implies d
X
j
0 mod ð
X
k
; 1 k m=2Þ, and in that case
X
j
¼
0; j ¼ 1; ...; m=2 defines an analytic submanifold of M, the complexified tangent
spaces to which are T
ðx;yÞ
ðMÞ
C
and in which hence m=2 analytic complex coor-
dinates z
j
may be chosen. Since fX
j
;
X
j
g is a base in T
ðx;yÞ
ðMÞC, the exterior
differential 2-form dX
j
may be written as
dX
j
¼
X
k\l
A
j
kl
X
k
^ X
l
þ
X
kl
B
j
kl
X
j
^
X
l
þ
X
k\l
C
j
kl
X
k
^
X
l
;
and (9.115) means C
j
kl
0. Since these expressions for dX
j
are tensor relations,
C
j
kl
is a tensor and its vanishing and hence the condition (9.115) is invariant under
linear transformations of the X
j
as it must be. If an almost complex manifold
fulfills the condition (9.115), it is called integrable.
It has been proved by quite technical means that real analyticity need not be
presupposed, so that
342 9 Riemannian Geometry
every (smooth) integrable almost complex manifold M derives its almost
complex structure from a canonical complex structure of a complex manifold
structure of M.
Since for m ¼ 2 the condition (9.115) imposes no restriction (X ¼ dx þidy in
that case and hence dX ¼ 0), a two-dimensional almost complex manifold always
derives from a one-dimensional complex manifold, that is, from a Riemannian
surface. For all even m [ 2 there exist almost complex manifolds which cannot be
given the structure of a complex manifold. (This holds for instance for the real
sphere S
4
.)
Let M be a real manifold of even dimension m and let an almost complex
structure J on M be given. Let ðx
1
; ...; x
m
Þ be arbitrary local coordinates in M.In
view of (9.111), hJ
x
ðdx
j
Þ; o=ox
k
i¼hdx
j
; J
x
ðo=ox
k
Þi ¼
P
m
l
J
l
k
ðxÞhdx
j
; o=ox
l
i¼
J
j
k
ðxÞ. Hence, J
x
ðdx
j
Þ¼
P
m
k
J
j
k
ðxÞdx
k
. Now,
P
m
k
J
j
k
ðxÞþid
j
k
dx
k
¼ðJ
x
þ iÞdx
j
and J
x
ðJ
x
þ iÞdx
j
¼ iði þ J
x
Þdx
j
; J
x
ðJ
x
iÞdx
j
¼iði þ J
x
Þdx
j
. This proves that
X
m
k
J
j
k
ðxÞþid
j
k
dx
k
is an exterior differential (1,0)-form, and
X
m
k
J
j
k
ðxÞid
j
k
dx
k
is an exterior differential (0,1)-form.
Suppose the integrability condition (9.115) is valid, It follows that
d
X
m
k
J
j
k
ðxÞþid
j
k
dx
k
!
¼
X
m
l\k
J
j
kl
dx
l
^ dx
k
0 mod
X
m
p
J
n
p
ðxÞþid
n
p
dx
p
; 1 n m
!
;
where
J
j
kl
¼
oJ
j
k
ox
l
oJ
j
l
ox
k
: ð9:116Þ
The 1-forms
P
m
p
ðJ
n
p
ðxÞþid
n
p
Þdx
p
; 1 p m; of type ð1; 0Þ annihilate the
subspace of tangent vectors
P
m
n
ðJ
n
p
ðxÞid
n
p
Þo=ox
n
; 1 p m; of type ð0; 1Þ.
Hence, the conditions above (9.116) can be expressed as
X
m
kl
J
j
kl
J
k
p
id
k
p
J
l
q
id
l
q
¼ 0; for all 1 j; p; q m;
or, with (9.116) and the second relation (9.111),
T
j
kl
¼ 0; T
j
kl
¼
X
m
p
J
j
kp
J
p
l
J
j
lp
J
p
k
: ð9:117Þ
9.7 Complex, Hermitian and Kählerian Manifolds 343
The tensor field T
j
kl
of type ð1; 2Þ is called the torsion tensor field of the almost
complex structure J (not to be confused with the torsion tensor of a linear
connection).
The almost complex structure J on M is integrable, iff its torsion tensor field is
zero.
A very systematic complex external calculus can be developed on an integrable
almost complex manifold (and hence on a complex manifold). If the components
C
j
1
...j
p
;
j
1
...
j
q
of (9.104) are smooth functions on M, then (9.104) is called an exte-
rior differential ðp; qÞ-form. Like in the real case, the algebra of all complex
exterior forms over the ring of smooth complex functions on M may be considered,
of which the exterior ðp; qÞ-forms are homogeneous elements. Clearly, like in the
case of the 1-form X
j
,ifX is an exterior ðp; qÞ-form, then dX is a sum of a
ðp þ 2; q 1Þ-form (in general non-zero, if q [ 0), a ðp þ 1; qÞ-form, a ðp; q þ1Þ-
form and a ðp 1; q þ2Þ-form (in general non-zero, if p [0). Let oX be the part
of dX which is a ðp þ 1; qÞ-form and let
oX be the part of dX which is a ðp; q þ1Þ-
form. Take the coordinate representation (9.104)ofX and consider the integra-
bility condition (9.115). It is easily seen that either of the conditions
d ¼ o þ
o;
o
2
¼ 0 ð9:118Þ
is equivalent to the integrability condition (9.115).
Proof Indeed, in either case (9.115) and (9.118), left condition, the ðp þ 2;
q 1Þ-part and the ðp 1; q þ2Þ-part of dX vanish, and the other two parts are
obtained under both conditions. Moreover, d
2
¼ 0 which holds generally implies
o
2
þ o
o þ
o o þ
o
2
¼ 0 because of the left condition of (9.118), and, since
forms of different type are linearly independent, o
2
¼ 0; o
o þ
o o ¼ 0;
o
2
¼ 0.
Conversely, for a smooth function F one obtains o F ¼
P
j
F
j
X
j
with smooth
functions F
j
. Now,
o
2
F is the ð0; 2Þ-part of dð
oFÞ¼dðd oÞF ¼dðoFÞ¼
P
j
ðdF
j
Þ^X
j
P
j
F
j
dX
j
. The first term has no ð0; 2Þ-part, and that of the
second term is ð1=2Þ
P
jkl
F
j
C
j
kl
X
k
^
X
l
. Hence,
o
2
¼ 0 implies C
j
kl
¼ 0 which is
equivalent to (9.115). h
With dz
j
¼ dx
j
þ idy
j
and (9.114), the differential of a smooth complex
function F on M may be written as
dF ¼
X
m=2
j¼1
oF
ox
j
dx
j
þ
oF
oy
j
dy
j
¼
X
m=2
j¼1
oF
oz
j
dz
j
þ
oF
oz
j
dz
j
;
hence
oF ¼
X
m=2
j¼1
oF
oz
j
dz
j
;
oF ¼
X
m=2
j¼1
oF
oz
j
dz
j
: ð9:119Þ
344 9 Riemannian Geometry
For a general exterior differential ðp; qÞ-form
X ¼
X
k
1
\\k
p
;
l
1
\\
l
q
X
k
1
...k
p
;
l
1
...
l
q
dz
k
1
^^dz
k
p
^ dz
l
1
^^dz
l
q
ð9:120Þ
one has (cf. (3.40))
oX ¼
X
pþ1
r¼1
ð1Þ
rþ1
X
k
1
\\k
pþ1
;
l
1
\\
l
q
oX
k
1
...k
r1
k
rþ1
...k
pþ1
;
l
1
...
l
q
oz
k
r
dz
k
1
^ ...^ dz
k
p
^ dz
l
1
^ ...^ dz
l
q
ð9:121Þ
and
oX ¼
X
qþ1
s¼1
ð1Þ
pþsþ1
X
k
1
\\k
p
;
l
1
\\
l
qþ1
oX
k
1
...k
p
;
l
1
...
l
s1
l
sþ1
...
l
qþ1
oz
l
s
dz
k
1
^^dz
k
p
^ dz
l
1
^^dz
l
qþ1
:
ð9:122Þ
Splitting F in (9.119) into its real and imaginary parts as in (9.97) it is readily
seen that
F is analytic, iff
oF ¼ 0:
Likewise, if X is an exterior differential ðp; 0Þ-form, it is analytic, iff
oX ¼ 0,
and then it holds that dX ¼ oX.
If M is a complex manifold of (complex) dimension n and ðz
1
; ...; z
n
Þ is a local
coordinate system, then the tangent vectors o=oz
j
span the (complex) tangent
space T
z
ðMÞ. If a Hermitian structure is given on T
z
ðMÞ for every z 2 M so that in
every coordinate neighborhood the functions H
jk
ðzÞ¼Hððo=oz
j
Þ; ðo=oz
k
ÞÞ are
smooth functions of z, then M is called a Hermitian manifold. If moreover, the
Kählerian form G of H, in local coordinates
iG ¼
1
2
X
n
j;k¼1
H
jk
dz
j
^ dz
k
; ð9:123Þ
where G is a real-valued differential ð1; 1Þ-form, is a closed differential form, that
is, dG ¼ 0, then M is called a Kählerian manifold.
The definition of bundles and connections transfers readily to complex
manifolds M. If the typical fiber of a fiber bundle is a complex vector space V of
dimension n, a complex vector bundle on M is obtained. Complex tangent and
cotangent bundles, as well as tensor and exterior bundles on the basis of the
9.7 Complex, Hermitian and Kählerian Manifolds 345
former, are the constructions in which the so far considered tangent, cotangent
and tensor fields and complex exterior differential forms live. In particular, the
consideration of complex frame fields leads to linear connections, for which the
notion of torsion and curvature forms can be generalized. Complex bundles were
occassionally already discussed as complex Lie groups and their Lie algebras or
in connection with characteristic classes. If a positive Hermitian structure is
smoothly assigned to the fibers of a complex vector bundle, then it is called a
Hermitian vector bundle. It turns out that a Hermitian manifold is Kählerian, iff
its Hermitian tangent bundle is torsion-free. For further reading see for instance
[7]or[3].
References
1. Warner, F.W.: Foundations of Differentiable Manifolds and Lie Groups. Springer, New York
(1983)
2. Fuchs, J.: Affine Lie Algebras and Quantum Groups. Cambridge University Press, Cambridge
(1992)
3. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, Interscience, vol. I and II.
New York (1963, 1969)
4. Landau, L.D., Lifshits, E.M.: Classical Theory of Fields. Pergamon Press, London (1989)
5. t’Hooft, G. (eds.): 50 Years of Yang–Mills Theory. World Scientific, Singapore (2005)
6. Scolarici, G., Solombrino, L.: On the pseudo-Hermitian nondiagonalizable Hamiltonians.
J Math Phys 44, 4450–4459 (2003)
7. Chern, S.S., Chen, W.H., Lam, K.S.: Lectures on Differential Geometry. World Scientific,
Singapore (2000)
346 9 Riemannian Geometry