## On three-dimensional trans-Sasakian manifolds.(English)Zbl 1175.53058

Let $$M$$ be a three-dimensional (connected) trans-Sasakian manifold of type $$(\alpha,\beta)$$ with $$\alpha$$, $$\beta$$ constants. In this paper the authors prove that the following properties are equivalent:
(1) $$M$$ is locally $$\phi$$-symmetric;
(2) $$M$$ has $$\eta$$-parallel Ricci tensor;
(3) $$M$$ has constant scalar curvature.
An example of a three-dimensional locally $$\phi$$-symmetric trans-Sasakian of type $$(0, 1)$$ is given.
Reviewer: D. Perrone (Lecce)

### MSC:

 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)