On three-dimensional conformally flat quasi-Sasakian manifolds

On three dimensional quasi-Sasakian manifolds

“…Now by Theorem 3.6 of [6], such a space is conformally flat with constant scalar curvature. Consequently, if R,S denote the curvature tensor and the Ricci tensor of M, then…”

“…In this paper, we have considered a 3-dimensional quasi-Sasakain manifold. Olszak [6] proved that such a space is conformally flat with constant scalar curvature and hence the structure function β is constant. In this paper we have shown that in a 3-dimensional quasi-Sasakian manifold with constant scalar curvature, a second order symmetric parallel tensor is a constant multiple of the associated metric tensor.…”

“…Olszak [6] proved that M is conformally flat with constant scalar curvature and hence its structure function β is constant. We have shown that in such M, a second order symmetric parallel tensor is a constant multiple of the associated metric tensor.…”

On Three-Dimensional Locally Φ-Recurrent Quasi-Sasakian Manifolds

“…On a 3-dimensional quasi-Sasakian manifold, the structure function f3 was defined by Z. Olszak [4] and with the help of this function he has obtained necessary and sufficient conditions for the manifold to be conformally flat [5]. Next he has proved that if the manifold is additionally conformally flat with ¡3 = constant, then (a) the manifold is locally a product of R and a 2-dimensional Kahlerian space of constant Gauss curvature (the cosympletic case), or, (b) the manifold is of constant positive curvature (the non-cosympletic case, here the quasi-Sasakian structure is homothetic to a Sasakian structure).…”

“…Next he has proved that if the manifold is additionally conformally flat with ¡3 = constant, then (a) the manifold is locally a product of R and a 2-dimensional Kahlerian space of constant Gauss curvature (the cosympletic case), or, (b) the manifold is of constant positive curvature (the non-cosympletic case, here the quasi-Sasakian structure is homothetic to a Sasakian structure). An example of a three-dimensional quasi-Sasakian structure being conformally flat with non-constant structure function is also described in [5].…”