On a Generalized Quarter-Symmetric J-Connection in an Almost Hermitian Manifold

We have introduced a generalized type of quarter-symmetric connection (namely, generalized quarter-symmetric J-connection) in an almost Hermitian manifold which unifies the different types of quarter-symmetric metric and non-metric connections in an almost Hermitian manifold. In this paper, we prove that an almost Hermitian manifold is almost Kähler (resp. Hermitian) if and only if it is almost Kähler with respect to a generalized quarter-symmetric J-connection (resp. Hermitian with respect to a generalized quarter-symmetric J-connection). As a consequence, we obtain that an almost Hermitian manifold is Kähler if and only if it is Kähler with respect to a generalized quarter-symmetric J-connection. Furthermore, some properties of almost analytic vector field with respect to a generalized quarter-symmetric J-connection are also given.