Integrability of almost Kaehler manifolds

An integrability theorem for almost Kähler manifolds

In this paper we prove a new geometric integrability theorem for almost complex manifolds. Let M be a connected, oriented, smooth 2m real dimensional manifold with an U(m) structure on its tangent bundle. Moreover let ∇ be a compatible connection on it. Assume the complexified curvature of the connection has vanishing (0, 2) component. Then we claim that M can be given the structure of an m com...

Lecture on Kaehler manifolds

An integrability theorem for almost complex manifolds I

In this paper we prove a new geometric integrability theorem for almost complex manifolds. Let M be a connected, smooth 2m real dimensional manifold with an U(m) structure on its tangent bundle and ∇ be a compatible connection on it. Assume the complexified curvature of the connection has vanishing (0, 2) component. Then we claim that M can be given the structure of an m complex dimensional Käh...

Ricci tensor for $GCR$-lightlike submanifolds of indefinite Kaehler manifolds

We obtain the expression of Ricci tensor for a $GCR$-lightlikesubmanifold of indefinite complex space form and discuss itsproperties on a totally geodesic $GCR$-lightlike submanifold of anindefinite complex space form. Moreover, we have proved that everyproper totally umbilical $GCR$-lightlike submanifold of anindefinite Kaehler manifold is a totally geodesic $GCR$-lightlikesubmanifold.

The Curvature and the Integrability of Almost-kähler Manifolds: a Survey

Situated at the intersection of Riemannian, symplectic and complex geometry, Kähler geometry is a subject that has been intensively studied in the past halfcentury. The overlap of just Riemannian and symplectic geometry is thewider area of almost-Kähler geometry. Although problems specific to almost-Kähler geometry have been considered in the past (see e.g. [42, 44, 77]), it is fair to say that...