# zbMATH — the first resource for mathematics

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