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Homogeneous and \(h\)-contact unit tangent sphere bundles. (English) Zbl 1195.53045

Let \((M,g)\) denote a Riemannian manifold. A Riemannian \(g\)-natural metric \(G\) on \(TM\) is said to be of Kaluza-Klein type if and only if horizontal and vertical distributions are \(G\)-orthogonal. This class of metrics includes the Sasakian and Cheeger-Gromoll metrics. By investigating metrics of Kaluza-Klein type and associated contact metric structures on \(T_1M\) the authors obtain new characterizations of two-point homogeneous and \(H\)-contact spaces in terms of geometric propertis of the tangent sphere bundle. The investigations are focussed on the following questions: If \(T_1M\) is homogeneous, is \((M,g)\) necessarily two-point homogeneous? If \(T_1M\) is \(H\)-contact is \((M,g)\) Einstein?

MSC:

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