Yunhee Euh

A generalization of a 4-dimensional Einstein manifold

A weakly Einstein manifold is a natural generalization of a 4-dimensional Einstein manifold. In this paper, we shall give a characterization of a weakly Einstein manifold in terms of so-called generalized Singer-Thorpe bases. As an application, we prove a generalization of the Hitchin inequality for compact weakly Einstein 4-manifolds. Examples are provided to illustrate the theorems.

A REMARK CONCERNING UNIVERSAL CURVATURE IDENTITIES ON 4-DIMENSIONAL RIEMANNIAN MANIFOLDS

We shall prove the universality of the curvature identity for the 4-dimensional Riemannian manifold using a different method than that used by Gilkey, Park, and Sekigawa \cite{GPS}.

ORTHOGONAL ALMOST COMPLEX STRUCTURES ON THE RIEMANNIAN PRODUCTS OF EVEN-DIMENSIONAL ROUND SPHERES

Abstract. We discuss the integrability of orthogonal almost complexstructures on Riemannian products of even-dimensional round spheresand give a partial answer to the question raised by E. Calabi concerningthe existence of complex structures on a product manifold of a round2-sphere and of a round 4-sphere. 1. IntroductionIt is well-known that a 2n-dimensional sphere S 2n admits an almost complexstructure if and only if n = 1 or 3 and that any almost complex structure onS 2 is integrable. Also, the complex structure on S 2 is unique with respect tothe conformal structure on it. A 2-dimensional sphere S 2 equipped with thiscomplex structure is biholomorphic to a complex projective line CP 1 . On thecontrary, it is a long-standing open problem whether S 6 admits an integrablealmost complex structure (namely, a complex structure) or not. Lebrun [4]gave a partial answer to this problem, that is, proved that any orthogonal al-most complex structure on a round 6-sphere is never integrable (see also [6],Corollary 5.2). On one hand, Sutherland proved that a connected product ofeven-dimensional spheres admits an almost complex structure if and only ifit is a product of copies of S

NOTES ON A QUESTION RAISED BY E. CALABI

We show that any orthogonal almost complex structure on a warped product Riemannian manifold of an oriented closed surface with nonnegative Gaussian curvature and a round 4-sphere is never integrable. This provides a partial answer to a question raised by E. Calabi.