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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.

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