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$$H$$-contact unit tangent sphere bundles. (English) Zbl 1140.53014
Let $$(M,g)$$ be a Riemannian manifold. Its unit tangent bundle $$T_1 M$$ has a natural contact metric structure $$(\xi,\eta,\Phi,\bar{g})$$. Following a previous paper of the author [Differ. Geom. Appl. 20, No. 3, 367-378 (2004; Zbl 1061.53028)] we say that a metric contact structure is $$H$$-contact if the Reeb vector field $$\xi$$ is harmonic. E. Boeckx and L. Vanhecke [Differ. Geom. Appl. 13, No. 1, 77–93 (2000; Zbl 0973.53053)] have proved that if $$M$$ is a two-point homogeneous space then $$T_1 M$$ is $$H$$-contact, and asked if the converse of this holds. This paper investigates this question, confirming in several cases.

##### MSC:
 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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