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**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.

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.

Reviewer: Z.Olszak (Wrocław)

### MSC:

53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |

53C35 | Differential geometry of symmetric spaces |