## A contact metric manifold satisfying a certain curvature condition.(English)Zbl 0849.53030

In his doctoral thesis [Generalizations of locally symmetric spaces and locally $$\phi$$-symmetric spaces, Niigata University (1991)], the author has introduced a class of contact metric manifolds $$M(\phi, \xi, \eta, g)$$ satisfying the following condition (C): $$\overline\nabla_{\dot \gamma} R)(\cdot, \dot\gamma) \dot\gamma= 0$$ for any unit $$\overline\nabla$$-geodesic $$\gamma$$, where $$R$$ is the curvature tensor of $$g$$ and $$\overline \nabla$$ is a linear connection such that the structure tensors $$\phi$$, $$\xi$$, $$\eta$$, $$g$$ are parallel.
In the present paper, the author continues the studies from his thesis and his previous work [Rend. Circ. Mat. Palermo, II. Ser. 43, No. 1, 141-160 (1994; Zbl 0817.53019)]. At first, he proves that a Sasakian manifold satisfies condition (C) if and only if it is locally $$\phi$$-symmetric. Next, a three-dimensional non-Sasakian contact metric manifold satisfying condition (C) is of constant sectional curvature. As a consequence, one has the following: Any simply connected and complete three-dimensional contact metric manifold satisfying condition (C) is isometric to the unit sphere $$S^3$$ or $$SU(2)$$ or the universal covering space of $$SL(2, \mathbb{R})$$ or the Heisenberg group $$H^2$$, each with a special left invariant metric, or the Euclidean space $$E^3$$. Moreover, if a contact metric manifold of dimension $$2n+ 1\geq 5$$ satisfies condition (C) and $$\xi$$ belongs to the $$k$$-nullity distribution, then $$M$$ is a Sasakian locally $$\phi$$-symmetric space or it is locally the product of a flat $$(n+ 1)$$-dimensional manifold and an $$n$$-dimensional manifold of constant sectional curvature equal to 4.

### MSC:

 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C35 Differential geometry of symmetric spaces

Zbl 0817.53019
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