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

Curvature identities on almost Hermitian manifolds and applications

“…In [1], there is a similar inequality as in the theorem above in integration form for almost Kähler manifolds.…”

“…It turns out that the converse is not true even when the manifold is almost Kähler. Indeed, there are examples of strictly almost Kähler manifolds (almost Kähler but not Kähler) with vanishing (2,0)-part of the curvature tensor for the canonical connection which is equivalent to that the curvature tensor for the Levi-Civita connection satisfies the third Gray identity by Corollary 3.7(see for example [1]). However, when some curvature conditions are imposed, the answer turns out to be affirmative.…”

“…However the full conjecture is still open. For other topics and progresses about the integrability of almost Kähler manifolds, one can consult the survey [1] written by Apostolov and Drǎghici and references there in.…”

“…For the second result above, one should note that without any curvature assumption, even for almost Kähler manifolds, the vanishing of (2,0)-part of the curvature tensor of the canonical connection does not imply integrability. One can find such kind of examples in [1]. Moreover, the assumption that the second Ricci curvature is quasi positive can not be relaxed to nonnegative.…”

Melting of the Euclidean Metric to Negative Scalar Curvature

“…Here ω and g determine J. One may refer to [2] for some knowledge of almost Kähler geometry needed in this section. In this geometry, for the canonical hermitian connection ∇ determined by J we have the corresponding hermitian scalar curvature s ∇ .…”