## Contact metric manifolds satisfying a nullity condition.(English)Zbl 0837.53038

The authors study contact metric manifolds $$M^{2n+ 1}(\varphi, \xi, \eta, g)$$ [for definitions see D. E. Blair, Contact manifolds in Riemannian geometry, Lect. Notes Math. 509, Springer (1976; Zbl 0319.53026)] for which the characteristic vector field $$\xi$$ belongs to the so-called $$(\kappa, \mu)$$-nullity distribution. This means that the curvature operator $$R(X, Y)$$ of the manifold satisfies the condition $R(X, Y)\xi= \kappa (\eta(Y)X- \eta(X) Y)+ \mu(\eta(Y) hX- \eta(X) hY),\tag{$$*$$}$ where $$\kappa$$, $$\mu$$ are constants and $$2h$$ is the Lie derivative of $$\varphi$$ in the direction $$\xi$$. It is proved that for $$\kappa< 1$$, the curvature $$R$$ is completely determined for such manifolds; in particular, they have constant scalar curvature. In the case of $$\dim M= 3$$, contact metric manifolds satisfying $$(*)$$ are either Sasakian or locally isometric to one of the following Lie groups: $$\text{SO}(3)$$, $$\text{SL}(2, R)$$, $$E(2)$$, $$E(1, 1)$$ with a left invariant metric. The standard contact metric structure of the tangent sphere bundle $$T_1 M$$ satisfies $$(*)$$ if and only if the base manifold is of constant sectional curvature.

### MSC:

 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)

Zbl 0319.53026
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### References:

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