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Pseudo-Einstein unit tangent sphere bundles. (English) Zbl 1420.53056
Summary: In the present paper, we study the pseudo-Hermitian almost CR structure of unit tangent sphere bundle $$T_{1}M$$ over a Riemannian manifold $$M$$. Then we prove that if the unit tangent sphere bundle $$T_{1}M$$ is pseudo-Einstein, that is, the pseudo-Hermitian Ricci tensor is proportional to the Levi form, then the base manifold $$M$$ is Einstein. Moreover, when $$\dim M = 3$$ or $$4$$, we prove that $$T_{1}M$$ is pseudo-Einstein if and only if $$M$$ is of constant curvature 1.

##### MSC:
 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53D10 Contact manifolds (general theory)
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