Kouei Sekigawa

Four-dimensional almost Kähler locally symmetric spaces

Abstract In the framework of studying the integrability of almost Kahler manifolds, we prove that a four-dimensional compact almost Kahler locally symmetric space is a Kahler manifold.

Four-dimensional almost Kähler Einstein manifolds

SummaryWe prove that a four-dimensional compact almost Kähler manifold which is Einsteinian, and Einsteinian is a Kähler manifold.

UNIVERSAL CURVATURE IDENTITIES III

We examine universal curvature identities for pseudo-Riemannian manifolds with boundary. We determine the Euler–Lagrange equations associated to the Chern–Gauss–Bonnet formula and show that they are given solely in terms of curvature and the second fundamental form and do not involve covariant derivatives, thus generalizing a conjecture of Berger to this context.

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.

NOTES ON CRITICAL ALMOST HERMITIAN STRUCTURES

We discuss the critical points of the functional F‚,µ(J,g) = R M (‚?+µ? ⁄ )dvg on the spaces of all almost Hermitian structures AH(M) with (‚,µ) 2 R 2 i (0,0), where ? and ? ⁄ being the scalar curvature and the ⁄-scalar curvature of (J,g), respectively. We shall give several characterizations of Kahler structure for some special classes of almost Hermitian manifolds, in terms of the critical points of the functionals F‚,µ(J,g) on AH(M). Further, we provide the almost Hermitian analogy of the Hilberts result.

A curvature identity on a 6-dimensional Riemannian manifold and its applications

We derive a curvature identity that holds on any 6-dimensional Riemannian manifold, from the Chern-Gauss-Bonnet theorem for a 6-dimensional closed Riemannian manifold. Moreover, some applications of the curvature identity are given. We also define a generalization of harmonic manifolds to study the Lichnerowicz conjecture for a harmonic manifold “a harmonic manifold is locally symmetric” and provide another proof of the Lichnerowicz conjecture refined by Ledger for the 4-dimensional case under a slightly more general setting.

η-EINSTEIN TANGENT SPHERE BUNDLES OF CONSTANT RADII

We study the geometric properties of the base manifold for the tangent sphere bundle of radius r satisfying the η-Einstein condition with the standard contact metric structure. One of the main theorems is that the tangent sphere bundle of the n(≥3)-dimensional locally symmetric space, equipped with the standard contact metric structure, is an η-Einstein manifold if and only if the base manifold is a space of constant sectional curvature or .

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

A REMARK ON H-CONTACT UNIT TANGENT SPHERE BUNDLES

We shall give some curvature conditions for the unit tangent sphere bundle of an n( 4)-dimensional Riemannian manifold to be H- contact. Furthermore, we provide an example illustrating Main Theorem.

Compact Hermitian surfaces of pointwise constant holomorphic sectional curvature

Let M = ( M, J, g ) be an almost Hermitian manifold and U ( M )the unit tangent bundle of M . Then the holomorphic sectional curvature H = H ( x ) can be regarded as a differentiable function on U ( M ). If the function H is constant along each fibre, then M is called a space of pointwise constant holomorphic sectional curvature. Especially, if H is constant on the whole U ( M ), then M is called a space of constant holomorphic sectional curvature. An almost Hermitian manifold with an integrable almost complex structure is called a Hermitian manifold. A real 4-dimensional Hermitian manifold is called a Hermitian surface. Hermitian surfaces of pointwise constant holomorphic sectional curvature have been studied by several authors (cf. [2], [3], [5], [6] and so on).