Comm. Korean Math. Soc. 14 (1999), No. 4, pp. 777{787
ON LOW DIMENSIONAL ALMOST
HERMITIAN C -MANIFOLDS
Un Kyu Kim
Abstract. In this paper we characterize 4 or 6 dimensional almost
Hermitian C -manifold as an RK -manifold which the Ricci tensor of
the manifold coincides with the -Ricci tensor of the manifold and
obtain sucient conditions for the almost Hermitian C -manifolds of
dimension 4 or 6 to be Kaehlerian.
1. Introduction
Curvature identities are keys to understanding the geometry of almost
Hermitian manifolds. In particular, the following curvature identities in almost Hermitian manifolds are studied by many authors ([1]; [5]; [6]; [7]; [8];
):
(1)
R(X; Y; Z; W ) = R(X; Y; JZ; JW );
(2)
R(X; Y; Z; W ) = R(JX; JY; Z; W ) + R(JX; Y; JZ; W )
+R(JX; Y; Z; JW );
(3)
R(X; Y; Z; W ) = R(JX; JY; JZ; JW ):
It is well known that (1) =)(2)=)(3). An almost Hermitian manifold satisfying (3) is called an RK - manifold ([2]; [5]): Recently, C. C.
Hsiung and B. Xiong ([3]; [2]) introduced a new class of almost Hermitian
manifolds dened by
(4)
R(JX; JY; Z; W ) + R(JY; JZ; X; W ) + R(JZ; JX; Y; W ) = 0:
Received February 28, 1999. Revised June 18, 1999.
1991 Mathematics Subject Classication: 53C55, 53B20.
Key words and phrases: almost Hermitian C -manifolds, curvature, Ricci tensor,
-Ricci tensor.
This research was supported by the Korea Research Foundation made in the program 1998-015-D00034.
778
Un Kyu Kim
They called the almost Hermitian manifolds (M; J; g) satisfying (4)
an almost Hermitian C -manifold and showed that (1) =)(4)=)(3)
([3]; [2]). The almost Hermitian manifolds satisfying (2) have been studied by many authors ([1]; [4]; [5]).
The purpose of the present paper is to obtain characterizations of 4
or 6 dimensional almost Hermitian C -manifolds and to obtain sucient
conditions for the almost Hermitian C -manifolds of dimension 4 or 6 to
be Kaehlerian.
The author would like to express his hearty thanks to Professor Bonnie
Xiong for her kind comments.
2. Preliminaries
Let M be a Riemannian 2n-manifold, and g; J; and R be repectively a
Riemannian metric tensor, the tensor of an almost complex structure, and
the Riemann curvature tensor, with repect to g; of M . And let curvature
tensor R of M be given by
R(X; Y; Z; W ) = g([r ; r ]Z ; r[ ] Z; W )
for X; Y; Z; W 2 (M ):
X
Y
X;Y
Let L denote the subclass of the class L of almost complex structures
(or manifolds) satisfying ( ). Then we have ([1]; [2])
(2.1)
L1 L2 L3 L:
C. C. Hsiung and B. Xiong ([3]) called the almost complex structure
J satisfying
(2.2)
J J R + J J R + J J R = 0;
or equivalently (4), an almost C -structure, and corresponding manifold
an almost C -manifold. Also they proved ([3]) that the class C of all
almost C -structrues has the following inclusion relations with the classes
i
i
j
i
i
i1
L
j
i2
i
ij i3 k
i2
j
i3
i
ij i1 k
i3
j
i1
i
(2.3)
L1 C L 3 ; L 2 \ C = L 1 :
ij i2 k
On low dimensional almost Hermitian C -manifolds
779
By combining (2.1) and (2.3) we have
L1
(2.4)
L2
C
L3 L:
Let M be a dierentiable 2n-manifold of class C 1 with a Riemannian metric g and an almost complex structure J . If the tensor J further satises g(JX; JY ) = g(X; Y ) for X; Y 2 (M ), then J is said
to be an almost Hermitian structure with a Hermitian metric g on M ,
and the manifold is said to be almost Hermitian. The almost Hermitian structure J is an almost Kaehlerian structure if g((r J )Y; Z ) +
g((r J )Z; X ) + g((r J )X; Y ) = 0 holds and is said to be a Kaehlerian
structure if (r J )Y = 0 holds.
We denote the class of almost Hermitian L manifolds by AH for
i = 1; 2; 3; and Kaehlerian manifolds, almost Hermitian C -manifold and
almost Hermitian manifolds respectively by K; AHC , and AH . From
(2.4) we obtain the following inclusion relations among almost Hermitian
manifolds:
X
Y
Z
X
i
(2:5)
AH1
K
AH2
AHC
i
AH3 AH:
Let M = (M; J; g) be an almost Hermitian manifold. Then we have
(r J )JY = ;J (r J )Y; g((r J )Y; Z ) = ;g((Y; (r J )Z );
g((r J )Y; Y ) = 0; g((r J )Y; JY ) = 0
X
X
X
X
X
X
for X; Y; Z 2 (M ). The Ricci -tensor and the -scalar curvature
are dened respectively by
(X; Y ) = g(QX; Y ) = trace(Z 7! R(X; JZ )JY );
= trace Q
for all X; Y; Z 2 T (M ); p 2 M: An 2n-dimensional almost Hermitian manifold M is said to be a weakly -Einstein manifold if =
( =2n)g holds.
p
780
Un Kyu Kim
We dene three linear operators D ; i = 1; 2; 3 as the following:
(D1R)(X; Y; Z; W ) = 1 [R(JX; JY; Z; W ) + R(Y; JZ; JX; W )
2
+R(JZ; X; JY; W )];
(D2R)(X; Y; Z; W ) = 1 [R(X; Y; Z; W ) + R(JX; JY; Z; W )
2
+R(JX; Y; JZ; W ) + R(JX; Y; Z; JW )];
(D3R)(X; Y; Z; W ) = R(JX; JY; JZ; JW )
for all X; Y; Z; W 2 (M ):
i
3. 4-dimensional Almost Hermitian C -manifolds
In this section we shall investigate 4-dimensional almost Hermitian C manifolds. F. Tricerri and L. Vanhecke proved the following theorem.
Theorem A ([8]). Let M be an almost Hermitian manifold with real
dimension 4, curvature tensor R, Ricci tensor and scalar curvature .
Then we have the following identities:
(i) (I ; D1 )(I + D2 )(I + D3 )R = ; 41 ( ; )(31; 2);
(ii) (R + D3 R) ; (R + D 3 R) = 12 ( ; )g ;
where the tensor 1 and 2 are dened by
1 (X; Y )Z = g(X; Z )Y ; g(Y; Z )X;
2 (X; Y )Z = 2g(JX; Y )JZ + g(JX; Z )JY ; g(JY; Z )JX
for all X; Y; Z 2 (M ).
Let M be a 4-dimensional almost Hermitian manifold satisfying the
condition (3), that is D3 R = R. Then we have, from Theorem A,
(3.1) 3R(X; Y; Z; W ) ; R(JX; JY; Z; W ) + R(JX; Y; JZ; W )
+R(JX; Y; Z; JW ) ; 2R(Y; JZ; JX; W ) ; 2R(JZ; X; JY; W )
= ; 1 ( ; )[3g(X; Z )g(Y; W ) ; 3g(Y; Z )g(X; W )
4
;2g(JX; Y )g(JZ; W ) ; g(JX; Z )g(JY; W ) + g(JY; Z )g(JX; W )]:
On low dimensional almost Hermitian C -manifolds
781
Using the rst Bianchi identity, we obtain
(3.2) 3R(X; Y; Z; W ) ; R(JX; JY; Z; W ) + R(JX; Y; JZ; W )
+R(JX; Y; Z; JW ) ; 2R(Y; JZ; JX; W ) ; 2R(JZ; X; JY; W )
= 3R(X; Y; Z; W ) ; R(JX; JY; Z; W ) + R(JX; Y; JZ; W )
+R(JX; Y; Z; JW ) + 2[R(JZ; JX; Y; W ) + R(X; JY; JZ; W )
+R(JX; Y; JZ; W ) + R(JY; JZ; X; W )]:
Now suppose that (M; J; g) is a 4-dimensional almost Hermitian C manifold. Then we have, by the help of (4), (3.1) and (3.2),
(3.3) 3[R(X; Y; Z; W ) ; R(JX; JY; Z; W )
+R(JX; Y; JZ; W ) + R(JX; Y; Z; JW )]
= ; 1 ( ; )[3g(X; Z )g(Y; W ) ; 3g(Y; Z )g(X; W )
4
;2g(JX; Y )g(JZ; W ) ; g(JX; Z )g(JY; W ) + g(JY; Z )g(JX; W )]:
Let us take an orthonormal basis fe1 ; Je1 = e2; e3 ; Je3 = e4g. If we
put X = e1 , Y = e2; Z = e3 ; W = e4 in (3.3), we obtain = :
Conversely, we assume that (M; J; g) is a 4-dimensional almost Hermitian manifold satisfying (3) and = . Then (3.1) implies
(3.4) 3R(X; Y; Z; W ) = R(JX; JY; Z; W ) ; R(JX; Y; JZ; W )
;R(JX; Y; Z; JW ) + 2R(Y; JZ; JX; W )
+2R(JZ; X; JY; W );
which implies
(3.5) 3R(JX; JY; Z; W ) = R(X; Y; Z; W ) + R(X; JY; JZ; W )
+R(X; JY; Z; JW ) ; 2R(JY; JZ; X; W )
;2R(JZ; JX; Y; W ):
By the cyclic sum of (3.5) with respect to X; Y and Z , we obtain
R(JX; JY; Z; W ) + R(JY; JZ; X; W ) + R(JZ; JX; Y; W ) = 0;
where we have used the rst Bianchi identity and (3). Thus the manifold
M is an almost Hermitian C -manifold.
On the other hand, by the help of (ii) of Theorem A, we can easily see
that (3) and = hold if and only if (3) and = hold. Hence we
get the following characterization of the 4-dimensional almost Hermitian
C -manifolds.
782
Un Kyu Kim
Theorem 1. Let (M; J; g ) be a 4-dimensional almost Hermitian man-
ifold. Then M is an almost Hermitian C -manifold if and only if M satises
the condition (3) and = .
4. 6-dimensional Almost Hermitian C -manifolds
In this section we shall study 6-dimensional almost Hermitian C -manifolds.
For a (0, 2) type tensor S , we dene '(S ) and (S ) by
'(S )(X; Y; Z; W ) = g(X; Z )S (Y; W ) + g(Y; W )S (X; Z )
;g(X; W )S (Y; Z ) ; g(Y; Z )S (X; W );
and
(S )(X; Y; Z; W ) = 2g(X; JY )S (Z; JW ) + 2g(Z; JW )S (X; JY )
+g(X; JZ )S (Y; JW ) + g(Y; JW )S (X; JZ )
;g(X; JW )S (Y; JZ ) ; g(Y; JZ )S (X; JW )
respectively. F. Tricerri and L. Vanhecke proved the following theorem.
Theorem B ([8]). Let M be an almost Hermitian manifold with
dimension 6 and curvature tensor R. Then we have the following identity:
(I ; D1)(I + D2)(I + D3)R = ; 21 (3' ; )[(R + D3R) ; (R + D3R)]
+ 1 ( ; )(31 ; 2 );
4
where f(R + D3 R)g(X; Y ) = trace (Z 7! R(Z; X )Y ; JR(JZ; JX )JY );
f (R + D3R)g(X; Y ) = trace(Z 7! R(X; JZ )JY ; JR(JX; Z )Y ):
Let M be a 6-dimensional almost Hermitian manifold satisfying the
condition (3), that is D3 R = R. Then we obtain, from Theorem B,
(I ; D1 )(I + D2)(I + D3)R = ;(3' ; )( ; )+ 14 ( ; )(31 ; 2 );
which implies
On low dimensional almost Hermitian C -manifolds
783
(4.1) 3R(X; Y; Z; W ) ; R(JX; JY; Z; W ) + R(JX; Y; JZ; W )
+R(JX; Y; Z; JW ) ; 2R(Y; JZ; JX; W ) ; 2R(JZ; X; JY; W )
= ;3[g(X; Z )( ; )(Y; W ) + g(Y; W )( ; )(X; Z )
;g(X; W )( ; )(Y; Z ) ; g(Y; Z )( ; )(X; W )]
+2g(X; JY )( ; )(Z; JW ) + 2g(Z; JW )( ; )(X; JY )
+g(X; JZ )( ; )(Y; JW ) + g(Y; JW )( ; )(X; JZ )
;g(X; JW )( ; )(Y; JZ ) ; g(Y; JZ )( ; )(X; JW )
+ 41 ( ; )[3g(X; Z )g(Y; W ) ; 3g(Y; Z )g(X; W )
;2g(JX; Y )g(JZ; W ) ; g(JX; Z )g(JY; W ) + g(JY; Z )g(JX; W )]:
Now suppose that (M; J; g) is a 6-dimensional almost Hermitian C manifold. Then we have, by the help of (4), (3.2) and (4.1),
(4.2) 3[R(X; Y; Z; W ) ; R(JX; JY; Z; W )
+R(JX; Y; JZ; W ) + R(JX; Y; Z; JW )]
= ;3[g(X; Z )( ; )(Y; W ) + g(Y; W )( ; )(X; Z )
;g(X; W )( ; )(Y; Z ) ; g(Y; Z )( ; )(X; W )]
+2g(X; JY )( ; )(Z; JW ) + 2g(Z; JW )( ; )(X; JY )
+g(X; JZ )( ; )(Y; JW ) + g(Y; JW )( ; )(X; JZ )
;g(X; JW )( ; )(Y; JZ ) ; g(Y; JZ )( ; )(X; JW )
+ 41 ( ; )[3g(X; Z )g(Y; W ) ; 3g(Y; Z )g(X; W )
;2g(JX; Y )g(JZ; W ) ; g(JX; Z )g(JY; W )
+g(JY; Z )g(JX; W )]:
Let fe1; e2 = Je1 ; e3; e4 = Je3 ; e5; e6 = Je5 g be an orthonormal
basis. If we put X = e1; Y = e2 ; Z = e3; W = e4 in (4.2), then we have
(4.3)
(e1 ; e1 ) + (e3; e3 ) ; [ (e1 ; e1) + (e3; e3)] = 41 ( ; ):
Similarly, if we put X = e1 ; Y = e2 ; Z = e5 ; W = e6 and X = e3;
Y = e4; Z = e5 ; W = e6 in (4.2), respectively, then we have
784
Un Kyu Kim
(4.4) (e1; e1 ) + (e5 ; e5) ; [ (e1 ; e1) + (e5; e5 )] = 1 ( ; );
4
(4.5) (e3; e3 ) + (e5 ; e5) ; [ (e3 ; e3) + (e5; e5 )] = 14 ( ; );
respectively.
It is well known that (X; Y ) = (JX; JY ); (X; Y ) = (Y; X ) =
(JX; JY ) in an almost Hermitian manifold M satisfying (3), ([5],[8]).
Therefore we have, from (4.3), (4.4) and (4.5), ; = 34 ( ; ). And
hence we obtain = . From (4.3) and (4.4), we have
(e1 ; e1) + 2 ; (e1; e1 ) + 2 = 0;
which implies (e1 ; e1) = (e1; e1 ) since = . From (4.4) and (4.5)
we obtain (e3 ; e3) = (e3 ; e3) and (e5 ; e5) = (e5; e5). Thus we have
(e ; e ) = (e ; e ) for i = 1; 2; ; 6. Since (X; JX ) = 0 and (X; JX )
= 0, we have (e1 ; e2) = (e3 ; e4) = (e5 ; e6) = (e1 ; e2) = (e3; e4) =
(e5; e6 ) = 0.
i
i
i
i
If we put X = Z = e5; Y = e1 ; W = e3 in (4.2), we obtain
( ; )(e1 ; e3) = R(e5; e1 ; e3; e5) + R(e6 ; e1; e3; e6 ) ; R(e6 ; e5; e1; e4 )
= R(e5; e1 ; e3; e5) + R(e6 ; e1; e3; e6 ) + R(Je5; Je6 ; e1; e4)
= R(e5; e1 ; e3; e5) + R(e6 ; e1; e3; e6 ) ; R(Je6 ; Je1; e5 ; e4)
;R(Je1 ; Je5; e6 ; e4) = 0;
where we have used the rst Bianchi identity and (4). Hence we get
(e1 ; e3) = (e1 ; e3):
If we put X = Z = e5; Y = e1 ; W = e4 in (4.2), we obtain (e1 ; e4) =
(e1; e4 ):
If we put X = Z = e3; Y = e1 ; W = e5 in (4.2), we obtain (e1 ; e5) =
(e1; e5 ):
Similarly, we have (e1 ; e6) = (e1; e6 ), (e3; e5 ) = (e3 ; e5), and
(e3 ; e6) = (e3 ; e6). Thus we have (e ; e ) = (e ; e ), that is, = .
Conversely, we assume that (M; J; g) is a 6-dimensional almost Hermitian manifold satisfying (3) and = . Then (4.1) implies (3.4).
By the same arguments as the 4 dimensional case, we obtain (4) and
i
j
i
j
On low dimensional almost Hermitian C -manifolds
785
hence the manifold (M; J; g) is an almost Hermitian C -manifold. Thus we
have the following characterization of the 6-dimensional almost Hermitian
C -manifold.
Theorem 2. Let (M; J; g ) be a 6-dimensional almost Hermitian manifold. Then M is an almost Hermitian C -manifold if and only if M satises
the condition (3) and = :
5. Einstein Almost Hermitian C -manifolds
In this section we shall investigate sucient conditions for an almost
Hermitian C -manifold of dimension 4 or 6 to be a Kaehler manifold. Hsiung and Xiong proved the following theorem.
Theorem ([3]). Suppose that M is an almost Hermitian C -manifold
of real dimension 2n = 4 with nonzero constant holomorphic sectional
curvature at each point and satises
(5.1)
R rJ +R r J +R r J =0
where J is the almost Hermitian C -structure, R
is the Riemann
curvature tensor, and r is the covariant derivation, with respect to the
almost Hermitian metric g of J . Then M is almost Kaehlerian.
It is known that an almost Hermitian C -manifold with constant holomorphic sectional curvature at each point is an Einstein manifold ([2]). We
shall improve the above theorem in the case of 4 or 6 dimensional almost
Hermitian C -manifolds, that is, we shall obtain the following Theorem
3 under the condition \Einstein" which is weaker than the condition
\constant holomorphic sectional curvature at each point".
Let (M; J; g) be an Einstein almost Hermitian C -manifold of dimension
2n(2n = 4; 6). Then the Ricci tensor and the -Ricci tensor satisfy
(5.2)
= = (=2n)g :
If we put
(5.3)
H = (1=2)J R ;
then we have
(5.4) H = 12 J (;R ; R ) = 12 J (R ; R ) = J R :
ih
hij k
ih
l
hikl
ih
j
hilj
k
j
hij k
i
j
ij
i
ji
ji
ji
ji
ji
ih
kj
ihkj
ih
kj
ih
j ikh
ikj h
ih
ij kh
hj ki
ij kh
786
Un Kyu Kim
Using (5.4), we obtain
= J J R = ;J J R = ;H J ;
which implies with (5.2)
g = ;H J :
(5.5)
2n
Transvecting (5.5) with J and taking account of (5.3), we obtain
(5.6)
J R = 2H = ; n J :
ba
t
ji
t
bj ta
i
ab
t
atj b
i
jt
i
t
ji
jt
i
i
k
ab
abj k
jk
jk
Dierentiating (5.6) covariantly along M , we get
R r J + J r R = ; n r J ;
which implies, with the second Bianchi identity,
(5.7)
R rJ +R r J +R r J
= ; (r J + r J + r J ):
ab
abj k
ab
l
l
j kab
abkl
j
j
kl
ab
abj k
jk
ablj
k
ab
l
n
l
l
jk
k
ab
lj
From (5.7) and the fact that an almost Kaehler manifold with = is a Kaehler manifold ([9]), we have the following
Theorem 3. Let (M; J; g ) be an Einstein almost Hermitian C -manifold
of dimension 4 or 6. If M satises
R r J + R r J + R r J = 0 and 6= 0;
then M is a Kaehlerian manifold.
ab
abj k
l
ab
abkl
j
ab
ablj
References
k
[1] A. Gray and L. Vanhecke, Almost Hermitian manifolds with constant holomorphic
sectional curvature, Casopis pro pestovani matematiky, roc. 104 (1979), 170-179.
[2] C. C. Hsiung, Almost complex and complex structures, Series in Pure Math. Vol.
20, World Scientic, Singapore, 1995.
[3] C. C. Hsiung and B. Xiong, A new class of almost complex structures, Annali di
Matematica pura ed applicata (IV), Vol. CLXVIII (1995), 133-149.
[4] U. K. Kim, On six dimensional almost Hermitian manifolds with pointwise constant holomorphic sectional curvature, Nihonkai Mathematical Journal 6 (1995),
no. 2, 185-200.
[5] T. Sato, On some almost Hermitian manifolds with constant holomorphic sectional
curvature, Kyungpook Mathematical Journal 29 (1989), no. 1, 11-25.
On low dimensional almost Hermitian C -manifolds
787
[6] S. Sawaki and K. Sekigawa, Almost Hermitian manifolds with constant holomorphic sectional curvature, Journal of dierential geometry 9 (1974), 123-134.
[7] T. Tanno, Constancy of holomorphic sectional curvature in almost Hermitian manifolds, Kodai Math. Sem. Rep. 25 (1973), 190-201.
[8] F. Tricerri and L. Vanhecke, Curvature tensors on almost Hermitian manifolds,
Transactions of the American Mathematical Society 257 (1981), no. 2, 365-398.
[9] K. Yano, Dierential geometry on complex and almost complex spaces, Pergamon
Press, New York, 1965.
Department of Mathematics Education
Sungkyunkwan University
Seoul 110-745, Korea
E-mail : [email protected]