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The curvature tensor of \(g\)-natural metrics on unit tangent sphere bundles. (English) Zbl 1148.53018

Let \(T_1M\) denote the unit tangent sphere bundle of a Riemannian manifold. The authors compute the curvature tensor corresponding to a metric on \(T_1M\) induced by a Riemannian g-natural metric on \(TM\) (either the Sasaki metric or the Cheeger-Gromoll metric). To this end, making use of the Gauss equation, the authors show a relation between the curvature tensor on \(T_1M\) and the tangential component of the curvature tensor of \(TM\). This relation involves also the shape operator of \(T_1M\) in \(TM\) and the second fundamental form.

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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