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\(g\)-natural metrics: new horizons in the geometry of tangent bundles of Riemannian manifolds. (English) Zbl 1211.53042

Summary: Traditionally, the Riemannian geometry of tangent and unit tangent bundles was related to the Sasaki metric. The study of the relationship between the geometry of a manifold \((M,g)\) and that of its tangent bundle \(TM\) equipped with the Sasaki metric \(g^s\) had shown some kinds of rigidity. The concept of naturality allowed O. Kowalski and M. Sekizawa to introduce a wide class of metrics on \(TM\) naturally constructed from some classical and non-classical lifts of \(g\). This class contains the Sasaki metric as well as the well known Cheeger-Gromoll metric and the metrics of Oproiu-type.
We review some of the most interesting results, obtained recently, concerning the geometry of the tangent and the unit tangent bundles equipped with an arbitrary Riemannian \(g\)-natural metric.

MSC:

53B20 Local Riemannian geometry
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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