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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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