A note on almost kähler manifolds

For anyn ≥ 2, we give examples of almost Kähler conformally flat manifoldsM 2n which are not Kähler. We discuss the meaning of these examples in the context of the Goldberg conjecture on almost Kahler manifold

On pseudo-Riemannian manifolds with recurrent concircular curvature tensor

It is proved that every concircularly recurrent manifold must be necessarily a recurrent manifold.Comment: 5 page

On totally geodesic affine immersions into affine manifolds of recurrent projective curvature

In the previous paper [8], the author has studied totally geodesic affine immersions $f:(M, \nabla \ ) \to \ ( \bar M \, \bar \nabla \ )$ in the case when $( \bar M \, \bar \nabla \ )$ is an affine manifold of recurrent curvature. It is shown there that $(M, \nabla \ )$ is flat or of recurrent curvature. And if $f$ is additionally umbilical with non-zero shape tensor and dim $M \ge \ 3$, then $(M, \nabla \ )$ is locally projectively flat. In the presented paper, the investigation of totally geodesic affine immersions is continued. We assume that the ambient manifold $( \bar M \ , \bar \nabla \ )$ is of recurrent projective curvature. It is proved that if dim $M \ge \ 3$, then $(M, \nabla \ )$ is locally projectively flat or of recurrent projective curvature. Moreover, if $f$ is additionally umbilical with non-zero shape tensor, then $(M, \nabla \ )$ is locally projectivvely flat