Local models and integrability of certain almost Kähler 4-manifolds

Abstract

We classify, up to a local isometry, all non-Kähler almost Kähler 4-manifolds for which the fundamental 2-form is an eigenform of the Weyl tensor, and whose Ricci tensor is invariant with respect to the almost complex structure. Equivalently, such almost Kähler 4-manifolds satisfy the third curvature condition of A. Gray. We use our local classification to show that, in the compact case, the third curvature condition of Gray is equivalent to the integrability of the corresponding almost complex structure. (2000): 53B20, 53C25 Mathematics Subject Classification

The curvature and the integrability of almost-Kähler manifolds: A survey

“…Using the local structure established in Theorem 8 and some further global arguments, it was also shown in [6] that the answer is negative. , and that J agrees with the standard orientation on these spaces.…”

“…We can write down the relation corresponding to d R by specifying the Weitzenböck decomposition of T M -valued 2-forms for the particular section N . This is done in [34] (see also [6]). …”

“…Four-dimensional examples. The following construction is discussed in [5,6] and generalizes the examples described in [73] and [13].…”

“…), the holomorphic function h is given by h = x + iy; in this case (14)(15) is homogeneous and isometric to the four-dimensional proper 3-symmetric space described in Example 3, see [5]. We also note here that an important feature of the above construction comes from the following observation: at any point where dw = 0, the Kähler nullity D = {T M ∋ X : ∇ X J = 0} of J is a two dimensional subspace of T M , which is tangent to the surface Σ.…”

Relating the curvature tensor and the complex Jacobi operator of an almost Hermitian manifold

“…For example, one says that C is nearly Kähler if (∇ x J)x = 0 for all tangent vectors x; we refer to [9] for further information concerning this class of manifolds. We say that C is almost Kähler if the two form Ω(x, y) := Jx, y is closed; we refer to [3] for a survey and to [2], [8] and [11] for some recent results concerning this class of manifolds.…”