4-dimensional anti-Kahler manifolds

Acta Math. Hungar. 104 (3) (2004), 265–269.

¨ 4-DIMENSIONAL ANTI-KAHLER MANIFOLDS H. KIM and J. KIM (Seoul)∗

Abstract. We show that every 4-dimensional anti-K¨ ahler manifold is Einstein and locally symmetric. In particular any 4-dimensional anti-K¨ ahler manifold with zero scalar curvature is flat.

1. Introduction An anti-K¨ahler manifold means a triple (M 2n , J, g) which consists of a smooth manifold M 2n of dimension 2n, an almost complex structure J and a metric g such that 5J = 0 where 5 is the Levi-Civita connection of g and the metric g is anti-Hermitian g(JX, JY ) = −g(X, Y ) for all vector fields X and Y on M 2n . Then the metric g has necessarily a neutral signature (n, n), M 2n is a complex manifold and there exists a holomorphic metric on M 2n [2]. This fact gives us some topological obstructions to an anti-K¨ahler manifold such that all its odd Chern numbers vanish because its holomorphic metric gives us a complex isomorphism between the complex tangent bundle and its dual and such that a compact simply connected K¨ahler manifold cannot be anti-K¨ahler because it does not admit a holomorphic metric. We extend J, g and the Levi-Civita connection 5 by C-linearity to the complexified tangent bundle T M C = T M ⊗ C. From now on we will use the same notations J, g and 5 for the complex extended almost complex structure of J, the complex extended metric of g and the complex extended Levi-Civita connection of 5 respectively. We will also use the same word the anti-K¨ahler manifold for the anti-K¨ahler manifold considered as a complex manifold with the complex extended g and J. The purpose of this note is to see the following results.

∗ The

authors wish to acknowledge the financial support of the Korea Research Foundation made in the program year of 1998. The second author was also supported by the Brain Korea 21 Project in 2003. Key words and phrases: 4-dimensional anti-K¨ ahler manifold, Einstein, locally symmetric, zero scalar curvature, flat. 1991 Mathematics Subject Classification: 32J27, 53B30, 53C25 53C56, 53C80.

c 2004 Akad´ 0236–5294/4/$ 20.00 emiai Kiad´ o, Budapest

266

H. KIM and J. KIM

Theorem 1. Every 4-dimensional anti-K¨ ahler manifold is Einstein. In particular any 4-dimensional anti-K¨ ahler manifold with zero scalar curvature must be flat. Theorem 2. Every 4-dimensional anti-K¨ ahler manifold is locally symmetric. We call an anti-K¨ahler manifold locally symmetric if 5R = 0. Here R is the curvature tensor of its complex extended metric g. In case of a 4dimensional K¨ahler manifold, the above facts do not in general happen. For instance CP 1 × T 2 with a product metric of the Fubini–Study metric and the flat one is not Einstein while it is a 4-dimensional K¨ahler manifold. Also a K3 surface with the Calabi–Yau metric is neither locally symmetric nor flat while it is a 4-dimensional K¨ahler manifold with zero scalar curvature.

2. Properties of anti-K¨ ahler manifolds Let (M 2m , J, g) be a 2m-dimensional anti-K¨ahler manifold. We extend J, g and the Levi-Civita connection of g by C-linearity to the complexification of the tangent bundle T M C = T M ⊗ C. Fix a (real) basis {X1 , . . . , Xm , JX1 , . . . , JXm } in each tangent space Tx M , then the set {Za , Za¯ } where Za = Xa − iJXa , Za¯ = Xa + iJXa forms a basis for each complexified tangent space Tx M ⊗ C. Unless otherwise stated, a, b, c, . . . run from 1 to m while A, B, C, . . . run through 1, . . . , m, ¯1, . . . , m. ¯ Then JZa = iZa and JZa¯ = −iZa¯ . We set gAB = g(ZA , ZB ) = gBA . Then the complex extended metric g satisfies ga¯b = g¯ba = 0 and gA¯B¯ = g¯AB . Conversely if the complex extended metric gAB satisfies the above properties then the initial metric must be anti-Hermitian. It will be customary to write a metric satisfying the above properties as (∗)

¯

ds2 = gab dz a dz b + ga¯¯b dz a¯ dz b .

In an adapted almost-complex coordinate, xµ = (xa , y a ≡ xm+a ), z a = xa + iy a , one has gµν dxµ dxν = 2 Re [gab dz a dz b ], where µ, ν = 1, . . . , 2m, a, b = 1, . . . , m and Re means real part. We define now the complex Christoffel symbols ΓC AB by (∗∗)

5ZA ZB = ΓC AB ZC .

If 5J = 0 then the torsion T and the Nijenhuis tensor N satisfy T (JX, JY ) = 21 N (X, Y ) for all vector fields X and Y . Since the complex extended LeviCivita connection 5 has vanishing torsion, the complex Christoffel symbols Acta Mathematica Hungarica 104, 2004

¨ 4-DIMENSIONAL ANTI-KAHLER MANIFOLDS

267

C are symmetric, i.e., ΓC AB = ΓBA . In this case the complex structure J is integrable so that the real manifold M 2m inherits the structure of a complex manifold. Let us now recall that there is a one to one correspondence between complex manifolds and real manifolds with an integrable complex structure. This means that there exists an atlas of real, adapted (local) coordinates (x1 , . . . , xm , y 1 , . . . , y m ) such that     ∂ ∂ ∂ ∂ = a, J = − a. J a a ∂x ∂y ∂y ∂x

We set Xa =

∂ , ∂xa

Za = Xa − iJXa = 2∂a ,

Za¯ = Xa + iJXa = 2∂a¯ ,

where ∂A = ∂z∂A , z a¯ = z¯a . It appears that (z a ) form an atlas of complex (analytic) coordinate charts on M . By using Christoffel formulas, one gets 1 CD ΓC (ZA gBD + ZB gDA − ZD gAB ) = g CD (∂A gBD + ∂B gDA − ∂D gAB ). AB = g 2 The curvature tensors R are then given (in complex coordinate) by the classical formula A A A E A E RBCD = ∂C ΓA BD − ∂D ΓBC + ΓEC ΓBD − ΓED ΓBC .

Theorem 3. Let (M 2m , J, g) be a 2m-dimensional anti-K¨ ahler manifold 1 m and (z , . . . , z ) an inherited (local) complex coordinate system. Then c (i) the (complex) Christoffel symbols satisfy ΓC AB = 0 except for Γab and ¯c ; Γca¯¯¯b = Γ ab (ii) the components of the complex extended metric gab are holomorphic functions; A a a ¯ = (iii) the curvature tensors satisfy RBCD = 0 except for Rbcd and R¯b¯ cd¯ a ¯ Rbcd . ¯

¯C Proof. From (∗∗) we have ΓC ¯B ¯ = ΓAB . Furthermore, since 5J = 0 the A connection satisfies the conditions 5ZA (JZB ) = J 5ZA ZB = i 5ZA ZB and 5ZA ZB¯ = J 5ZA ZB¯ = −i 5ZA ZB¯ . Acta Mathematica Hungarica 104, 2004

268

H. KIM and J. KIM

¯ = 0 for any B. In particular, And so this implies ΓaB¯c = ΓaBc

Γab¯c = g aD (∂b gc¯ D + ∂c¯gDb − ∂D gb¯c ) = g ad ∂c¯gbd = 0. So ∂c¯gbd = 0. By using both the definition of curvature tensor and the above described properties of Christoffel symbols, the described properties of curvature tensors are easily verified. This completes the proof. The Ricci tensor RAB and the scalar curvature S, respectively are given (in complex coordinates) as follows: by using the Einstein summaC A . Let (M 2m , g, J) be tion convention RAB = RACB and S = g AB RAB = RA an anti-K¨ahler manifold, i.e., a complex manifold of complex dimension m with a holomorphic metric gab (z), a, b = 1, . . . , m and a real metric gµν (x), µ, ν = 1, . . . , 2m defined by (∗). Theorem 4. The holomorphic metric gab (z) is Einstein if and only if the real metric gµν (x) is a solution of the Einstein equations. In other words S S we have Rab = 2m gab if and only if Rµν = 2m gµν . Proof. By Theorem 3 we have Ra¯b = 0. The (complex) Einstein equations RAB (g) = rgAB are thus equivalent to a pair of equations (∗∗∗)

Rab (gcd ) = rgab

and (∗∗∗∗).

Ra¯¯b (gc¯d¯) = rga¯¯b

To get a real solution of Einstein equations from (∗∗∗) and (∗∗∗∗), one uses then real coordinates (xµ ), µ = 1, . . . , 2m on M , i.e., z a = xa + ixm+a , a = 1, . . . , m and writes the real Ricci tensor Rµν in the from Rµν dxµ dxν ¯ = Rab dz a dz b + Ra¯¯b dz a¯ dz b . The result then follows. Proof of Theorems 1 and 2. Let {E1 , E2 } be a local unitary ba¯ 1212 and sis. Then we obtain R11 = R22 = R1212 and R¯1¯1 = R¯2¯2 = R¯1¯2¯1¯2 = R A a and RAB = 0 for the other cases because we have RBCD = 0 except for Rbcd a ¯ =R ¯ a . And hence these facts imply that every 4-dimensional antiR¯b¯ bcd cd¯ K¨ahler manifold is Einstein and that its curvature tensor R consists only of its scalar curvature which is constant [1]. This completes the proof of Theorems 1 and 2. Acknowledgements. The authors would like to thank Professor J. Szabados and referee for kind suggestions to lead this paper as better. Acta Mathematica Hungarica 104, 2004

¨ 4-DIMENSIONAL ANTI-KAHLER MANIFOLDS

269

References [1] A. L. Besse, Einstein Manifolds, Springer Verlag (1986). [2] A. Borowiec, M. Francaviglia and I. Volvovich, Anti-K¨ ahlerian manifolds, Differential Geometry and its Applications, 12 (2000), 281–289. (Received January 15, 2003) DEPARTMENT OF MATHEMATICS YONSEI UNIVERSITY SHINCHON 134 SEOUL KOREA E-MAIL: [email protected] [email protected]

Acta Mathematica Hungarica 104, 2004