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Characteristic reflections on unit tangent sphere bundles. (English) Zbl 0897.53010
The authors study some properties of unit tangent sphere bundles \(T_1M\) over Riemannian manifolds \(M\). Such a unit sphere bundle admits a natural contact metric structure if equipped with a Riemannian metric homothetic to the corresponding Sasaki metric on \(T_1M\).
Recall that a \(\varphi\)-symmetric space is, roughly speaking, a Sasakian manifold which admits a codimension one Riemannian submersion onto a Hermitian symmetric space. The main result of the paper says that the tangent sphere bundle \(T_1M\) is locally a \(\varphi\)-symmetric space (as a contact metric manifold) if and only if \(M\) is a space of constant curvature. If \(M\) is a space of constant curvature, some further geometric properties of \(T_1M\) are studied which are related to Ricci curvature and Jacobi operators.
The paper is partly of expository character.

MSC:
53B20 Local Riemannian geometry
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C35 Differential geometry of symmetric spaces
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