Abstract
The following question is considered: Which quasi-Sasakian (cosymplectic, Sasakian, or proper quasi-Sasakian) structures admit nontrivial concircular transformations of their metrics (i.e., determine Fialkow spaces), and under what conditions. It is proved that any cosymplectic manifold is a Fialkow space. Necessary and sufficient conditions for a Sasakian or a quasi-Sasakian manifold to be a Fialkow space are obtained. A fairly large class of Sasakian manifolds which are not Fialkow spaces is described.
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