ON
ALMOST-ANALYTIC VECTORS IN
ALMOST-KAHLERIAN MANIFOLDS
1
*
SHUN-ICHI
TACHIBANA
(Received
February
2, 1959)
In
pseudo-Kahlerian manifolds, many interesting results concerning con-
travariant or covariant pseudo-analytic vectors are known.
2)
Even though
there
were many papers about pseudo-Kahlerian manifolds, but were few
about almost-Kahlerian ones. Recently, M. Apte generalized Bochner's theorem
to
compact almost-Kahlerian manifolds. His work seems to be
very
suggestive
for me. In the present paper we shall generalize several theorems in pseudo-
Kahlerian manifolds to almost-Kahlerian ones. The main results are integral
formulas on vector
fields
in compact almost-Kahlerian manifolds.
In
§1 and §2 we shall prepare identities and lemmas and in §3 and §4 define
almost-analytic vectors which are generalizations of pseudo-analytic vectors.
As applications of integral formulas in §5, we shall obtain several theorems in
§6. In §7, we shall
give
a decomposition theorem of the Lie algebra of contra-
variant almost-analytic vectors in a compact almost-Kahler-Einstein manifold.
The
canonical connection
will
be introduced in §8 and in the last section, to
contravariant almost-analytic vectors, we shall generalize Apte's theorem.
1.
Identities.
In an ^-dimensional real differentiable manifold M with
local coordiantes \x
%
\
9
a tensor
field
φf such that
(i.
i)
ψrψl
= - δ/
is called an almost-complex structure. If an almost-complex structure φ/
and
a positive definite Riemannian metric tensor g
H
on M
satisfy
the relation
(1.
2)
9rs<Pj
r
<Pi
S
= 9n,
then
the pair (φf, ffjt) is called an almost-Hermitian structure. Then, from
(1.1) and (1. 2), we get
(1.
3) ψ
H
= — φ
ih
where ψa^ψI^Ίi' To an almost-Hermitian structure (φ/, g
H
), an exterior dif-
1)
This
paper
was
prepared
in a
term
when
the
present
author
was
ordered
to
study
at
Tohoku
University.
I
wish
to
express
my
sincere
thanks
to
Prof.
S. Sasaki for his en-
couragements
during
the
term.
2) For
example,
cf.
Yano,
K.
[7],
Lichnerowicz,
A. [3], Sasaki, S. and K.
Yano
[5],
Yano,
K.
and I. Mogi [9].
24:8
S. TACHIBANA
ferential form
φ =
φ
H
dx
3
Λ dx
ι
can be
assoicated.
An
almost-Hermitian
structure
is
called
an
almost-Kahlerian structure,
if
the
associated differential
form
φ is
closed,
and
then
the
manifold
M is
called
an
almost-Kahlerian
manifold.
Throughout
this paper,
by M we
shall
always
mean
an
^-dimensional
differentiable manifold with
a
fixed
almost-Kahlerian structure
(φf, g
H
). In
this section,
we
shall deduce identities which
are
useful
in
the
later sections.
In
our M,
the
form
φ
being closed,
so we
have
(1.
4) Vkψjh + V&ik + Viψkj = 0,
whέre
\7/t
denotes
the
operator
of
covariant derivative with respect
to the
Riemahnian
connection.
On
the
other hand, since
the
identity
3
V
r
<Prh
= —
~γ-
V[r<Pl*]<p
rP
<Ph
a
is
well
known
4)
,
in our
case,
we
have
(1.5)
V
r
φm
=0.
The
Nijenhuis' tensor
N
H
h
of
an
almost-complex structure
φ
}
1
is
defined
by
N
H
h
=
φKVι<Pi
h
-
Vi<Pι
h
)
~
<PiXVi<Pi
h
~
Vi<Pι
h
),
so
on
taking account
of
(1.
1),
(1.
3) and
(1.
4),
we
find
(1.
6) N
H
h
= 2
<pKVι<Pi
h
~
Vi<Pι
h
)
Let
R
k
ji
h
.
be
Riemannian curvature tensor,
that
is,
RkH
=
Ok\Jl\
— Oj\u\ + \kr\ \ji\ — \jr\ \ki\
where
3
fc
=
d/dx
k
,
and put
and
Rji
==
Rrji
}
Rkjih
~
Rkji
9rh
(1.7)
Rtj =
-γφ
M
Rp<irjφt
We notice
that,
in
pseudo-Kahlerian manifold,
Rtj = R
k
j
holds good.
The
Ricci's identities
are
given
by
the
following
formulas
for any
vector
field
Vi
and v\
Applying
to φf the
Ricci's identity,
we
have
3)
Indices
ί,j,
k,
P,q,r,s, run
over
1, , n.
Notations
are
followed
to
Yano,
K.
[7]
except
some
trivial
changes.
4) Schouten,
J. A. and K.
Yano
[6].
5)
For
example
Yano,
K. [7], p. 229
ON
ALMOST-ANALYTIC VECTORS
IN
ALMOST-KAHLERI
AN MANIFOLDS
2
Transvecting
the
last equation with
g
kι
and
using (1.
5), we
find
where we put φ
ir
=
φjg
vι
.
On the other hand, φ
ir
being
skew
symmetric
with respect to i and r, so we have
Hence,
it
follows
that
and
from which we obtain
Operating y
fc
= g
k
\
r
to (1. 4), we have
Hence
on taking account of (1.8) we find
(1.
9)
V
T
Vr<PH
=
φ
Pq
Rp*H
+
iίiVn
~
^<>rj
The
equations (1. 8) and (I. 9) are important identities in the later sec-
tions.
By a vector
field
v we
always
mean a contravariant vector
field
v\ a
covariant vector
field
v
t
=
g
ir
v
r
and a differential form ϋ = Vidx
1
. So the
word "a vector is closed" means that the corresponding form v is closed.
Let v be any vector
field,
then
from (1. 3) and Ricci's identity we get
(1.
10) *> ^
And from (1. 8) and (1.10) we obtain
(1.
11)
ψΐvVVrψH
= ~ v
r
R*
L
+ v
r
R
rl
,
(1.12) φιφ
ir
ViSJτVi = - vR% ,
where i?*
r
is
given
by (1. 7).
From
(1. 3) and (1. 4) we have
9»»V/(VV'V.' - *»W
=
-
φΛvV +
VV)
Interchanging r and i and the adding the equation thus obtained to the last
equation,
we get
W
+
vV>/ = - ψnWivV
+
vV
ω
V/
Consequently, for any vector
field
v the
following
equation holds good.
250 S.
TACHIBANA
(1.13)
Viv£V<p
n
+ vVW = - φ*W(W
r
Xv
h
φ" +
vV>Λ
2. Lemmas.
For
convenience sake,
we
shall expose several lemmas
which
are
well
known.
By V
n
we
shall
always
mean
an
^-dimensional
Rie-
mannian
manifold.
LEMMA
2.1.
6)
In a
compact,
orientable
V
n
, the
following
integral
for-
mulas
are
valid
for any
vector
field
v.
(2.1)
f
v
[(v
Γ
Vrf*
-RnvW
+
S(v)~\dσ
= 0,
(2.
2) J
Vn
[(v
r
Vrf,
+
RnV
τ
W
+
T(v)-]dσ
= 0,
where
dσ
means
the
volume
element
of the V
n9
and S(v) and T(v) are
defined
by
(2.
3)
S(V)
= 4^(VV - VV)
(VsV
r
- VrV
s
) + (v^r) (v'f Λ
(2.
4) T(v) = 4-(vV + V
r
f') (V.f
r
+ Vrv.) -
(V
r
v
r
)
(V
s
v.)
respectively.
In
a V
n
, a
vector field
v is
called
a
Killing vector
(or an
infinitesimal
isometry)
if it
satisfies
V
where
£
denotes
the
operator
of Lie
differentiation with respect
to v\ For
Killing vectors,
the
following theorem
is
well
known.
LEMMA
2. 2.
7)
In a
compact,
orientable
V
n
, a
necessary
and
sufficient
condition
for v to be a
Killing
vector
is
that
V
r
VrVi
4-
R
ri
v
r
= 0,
τj
r
v
r
= 0.
In
a V
n
, a
vector field
v is
called
a
conformal Killing vector
(or an in-
finitesimal
conformal
transformation)
if it
satisfies
f)
9H
= V&t + Vi^ = 2
φ&
t
,
.
where
Φ is a
scalar function. Then
as is
well
known,
it
holds
the
following
LEMMA
2.3.
8)
In a
compact,
orientable
V
n
, a
necessary
and
sufficient
condition
in
order
that
v be a
conformal
Killing
vector
is
that
6)
For
example, Yano,
K. [7],
p. 278.
7)
For
example, Yano, K.
[7],
p. 221.
8) Yano,K.
[7],
p. 278.
ON
ALMOST-ANALYTIC VECTORS IN
ALMOST-KAHLERI
AN MANIFOLDS 251
(2.
5)
V\rVi
+ RriV + ^—^
ViVrf
r
= 0.
n
In
a V
n
, a vector field v is called a protective Killing vector
9)
if it
satisfies
(2.
6)
$lji}
=
VjViV*
+ ^V = δ/ψi + δ
t
*ψ*
Ψi
being a certain vector field. Transvecting (2. 6) with g
jί
, we get
(2.
7)
v
r
VrV
h
+ iW = 2 ψ
Λ
By contraction with respect to h and i in (2. 6), it
follows
that
(2.
8)
VNrV
=(n
From
(2. 7) and (2. 8), we have
(2.
9)
V
r
Vr*>i
+ RriV = —
for a
projective
Killing
vector
v.
Since
an
almost-Kahlerian
manifold
M is an
orientable
Riemannian
manifold,
the
above
lemmas
and
arguments
are
valid
for our M.
For a
vector
field
v we
define
v by
(2.10)
Vi = ^ϋί, v* =
^
ίr
t;
r
= -
ψtv
u
then we
have
and
V^r^ =
(VVrtPi^Vt
+ 2 (V>Λ Vr^ί +
ψiVVrVt-
Substituting (1. 9) in the right hand side of the last equation, we get
from
which
we
find
(2. 11)
(V
r
VrV
t
~
i?
ri
?V
=
(V
Γ
VrVi
+
ΛnvV
-
2
i?
r
Vz/
+ 2 (vV)
(VrΨu)<P
P
V.
On the other hand, by
virtue
of (l. 2) and (1. 4), we
find
so
(2,11)
can be
written
as
(2. 12)
ίVVrVt
~ RnV'W =
(v'Vr^i
+
ΛriVV
9)
Yano,
K. [7], p. 133.