## On Lorentzian $$\alpha$$-Sasakian manifolds.(English)Zbl 1085.53023

Let $$M^{2n+1}$$ be a Lorentzian $$\alpha$$-Sasakian manifold. The authors prove that $$M$$ is locally isometric to a sphere $$S^{2n+1}(\alpha^2)$$ provided one of the following conditions holds: (1) $$M$$ is conformally flat; (2) $$M$$ is quasi-conformally flat; (3) $$R(X,Y)\cdot C = 0$$, where $$R$$ and $$C$$ are the curvature tensor and conformal curvature tensor of $$M$$, respectively.

### MSC:

 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)