On Einstein Hermitian manifolds

Hermitian structures on the product of Sasakian manifolds

“…Remark 3 From (3.34),we see easily check that the * -scalar curvatureτ * of the Einstein Hermitian manifold (S 2p+1 × S 2q+1 ,J 0, √ p q ,ḡ 0, √ p q ) (p = q, q > 1) is given byτ * = 4q(1 − p + q). Thus, taking account of the Remark 1, we see that there exists a 2ndimensional compact Einstein Hermitian manifold with constant * -scalar curvature for any integer n(≧ 3) which is not Kähler, and this completes the assertion of [6].…”

“…for X, Ȳ , Z ∈ X(M) [3,5,6]. A 2n-dimensional almost Hermitian manifold (M , J, ḡ) is called a weakly * -Einstein manifold if the equality ρ * = τ * 2n ḡ holds on M .…”

“…However, the situation concerning this fact is quite different in the 4-dimensional case. In fact, by applying the Riemannian version of the Goldberg-Sachs Theorem to a compact Einstein Hermitian surface M it follows that M is a locally conformal Kähler [1,6]. We may easily check that the Ricci * -tensor ρ * is symmetric in a locally conformal Kähler surface since the Lee form is closed ([3], (2.16)).…”

Geometric Realizations of Para-Hermitian Curvature Models

“…We refer to [2,4,45,59,75] for additional material on Hermitian geometry. We say an almost Hermitian manifold M = (M, g, J ) is Hermitian if J is an integrable almost complex structure, i.e.…”

Geometric realizations of Hermitian curvature models

“…Nevertheless, many fundamental questions remain unanswered. In this paper we shall work in the context of Hermitian geometry and refer to a few earlier works in the almost-Hermitian context [7,15,16,19] and in the Hermitian context [1,2,14,18,22]; the field is a vast one. Our main result, see Theorem 1.4 below, gives a complete answer to the fundamental question of when a curvature tensor in a Hermitian vector space is geometrically realizable by a Hermitian manifold.…”