Takashi Oguro

Four-Dimensional Almost Kähler Einstein and *-Einstein Manifolds

In the framework of studying the integrability of almost Kähler manifolds, we prove that a four-dimensional almost Kähler Einstein and -Einstein manifold is a Kähler manifold. Further, we estimate the *-scalar curvature of a four-dimensional compact almost Kähler Einstein and weakly *-Einstein manifold with negative scalar curvature.

Notes on the Goldberg Conjecture in Dimension Four

Concerning the integrability of almost Kahler manifolds, there is a longstanding conjecture by S.I. Goldberg, “A compact almost Kahler Einstein manifold is Kahler”. The conjecture is true in the case where the scalar curvature is non-negative. However, the conjecture is still open in the remaining case. In this note, we shall give a brief survey on the recent progress concerning the conjecture in four-dimensional case.

NOTES ON KAHLER SURFACES WITH DISTINCT CONSTANT RICCI EIGENVALUES

In this paper, we study a connected, simply connected homogeneous Kahler surface with distinct constant Ricci eigenvalues, and specify the local structure of them.