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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.

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