ON SOME COMPACT ALMOST KHLER LOCALLY SYMMETRIC SPACE

In the framework of studying the integrability of almost Kähler manifolds, we prove that if a compact almost Kähler locally symmetric space M is a weakly ,∗-Einstein vnanifold with non-negative ,∗-scalar curvature, then M is a Kähler manifold.


INTRODUCTION
An almost Hermitian manifold M (M, J, g) is called an almost Kghler manifold if the corresponding KRb_ler form is closed, or equivalently , g((VxJ)Y,Z) 0 for all X, Y, x,Y,X Z E 3(M), where X(M) denotes the Lie algebra of all smooth vector fields on M and x,Y,Z denotes the cyclic sum with respect to X, Y, Z.By the definition, a Kh'xler manifold (VJ 0)   is necessarily an almost Khahler manifold.It is well-known that if the almost complex structure of an almost KKb.ler manifold M is integrable, then M is a KRhler manifold.A non-KKhler, almost Khahler manifold is called a strictly almost Kghler manifold.Several examples of strictly almost Kiihler manifolds have been constructed ( [1], [6], [9], [13] and so on).
Concerning the integrability of almost KRhler manifolds, the following conjecture by Gold- berg is known ([4]).
CON3ECTURE.The almost complex structure of a compact almost KRhler Einstein manifold is integrable.
On any almost Hermitian manifold, we can define Pdcci ,-tensor, an analogue of the Ricci tensor, but involving the almost complex structure (see (2.1) below for the definition).On a Kghler manifold, the Ricci tensor and the Ricci ,-tensor coincide.Therefore, it is natural to consider star-version of the Goldberg conjecture.An almost Hermitian manifold is called weakly ,-Einstein if the Pdcci ,-tensor is a (not necessarily constant) multiple of the metric T. OGURO and ,-Einstein if the R.icci ,-tensor is a constant multiple of the metric.
In the present paper, concerning the star-version of the Goldberg conjecture, we shall prove the following.
THEOREM.Let M (M,J,g) be a compact almost KKhler locally symmetric space which is a weakly ,-Einstein manifold with non-negative ,-scalar curvature.Then, M is KKhler manifold.
The author express his sincere thanks to Prof. K. Sekigawa for his many valuable advice.

PRELIMINARIES
Let M (M, J, g) be a 2n-dimensional almost Hermitian manifold with the almost Hermit- ian structure (J, g) and the corresponding KKhler form of M defied by f/(X, Y) g(X, JY) for X, Y E 3(M).We assume that M is oriented by the volume form dM n! Let V be the Riemarmian connection and R its curvature tensor given by R(X,Y)Z [Vx,Vy]Z-V[x,y]Z for X, Y, Z E 3(M).We denote by p and r the associated Ricci tensor and the scalar curvature, respectively.Moreover, let p*, respectively r*, denote the Ricci .-tensor,respectively the .-scalarcurvature, defined by p*(x,y) g(Q*x,y) trace (z R(x, Jz)Jy), ( '* trace Q, for x, y, z T,M, the tangent space of M at p M. By using the first Bianchi identity, we have easily 1 p*(x,y) 5 trace (z R(x, Jy)Jz).
(2.2) Thus, in general, p* is neither symmetric nor skew-symmetric.But it satisfies the following identify.
Let el,..., e2,, } be an orthonormal basis of Tt, M at any point p q M. In the present paper, we shall adopt the following notational convention: and so on, where the latin indices run over the range 1, 2, 2n.Then, we have easily -s,, v,, -v,, v, -v,,,.Now, we assume that M (M, J,g) is an almost Khler manifold.Then it is known that M is a quasi-Kilaler manifold, namely, the equality V, Jik -VJ}k (2.4) is valid.Thus, it follows immediately that M is a semi-Khler manifold, namely, the equality a=l is valid.We now recall the following curvature identity established by Gray ([5]): a=l From (2.1) (2.3) and (2.6), we have easily (2.6) Further, we may easily observe that M is Khler if and only if B 0 holds identically on M.

PROOF OF THEOREM
Let M (M, J, g) be a 2n-dimensional compact almost Kler locally symmetric space which is weakly ,-Einstein with r* > 0. We define a smooth vector field on M by a=l i,3,k,l=l at each point p 6 M.Then, taking account of VR 0, the Ricci identity and (2.2), we have and further [IVJ]I 2(r" r). (2.7) On one hand, transvecting b(7bJ,)VbJ,t with (2.6) and taking account of (2.4), we have easily B=4A. (2.8) Since M is weakly .-Einstein,from (3.1), we have div A + 2r*(r* r).
Thus, form (2.7) and (2.8), we have finally Therefore, by Green's Theorem, we obtain a following integral formula: Since B _> 0 and r* > 0, it must follow that B 0 holds identically on M, and hence M is a Kgb_ler manifold.This completes the proof of Theorem.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning Now, we shall define smooth functions A, B on M respectively by A= E a,t,j,k,l=l 2n a,b,t,,k,l=l(2.3)at any point p 6 M.

•
Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation