## On the concircular curvature tensor of a contact metric manifold.(English)Zbl 1084.53039

Let $$(M, g)$$ be a Riemannian manifold and let $$R, C$$ denote its Riemannian and Weyl conformal curvature tensors respectively. $$(M, g)$$ is semi-symmetric if $$R(X,Y)\cdot R=0,\;\forall \;X, Y\in TM$$, where $$R(X,Y)$$ acts on $$R$$ as a derivation. Contact metric manifolds satisfying semi-symmetric type equations had been studied by several authors. For example, S. Tanno [Kodai Math. Sem. Rep. 21, 448–458 (1969; Zbl 0196.25501)] showed that a semi-symmetric $$K$$-contact metric manifold $$M^{2n+1}$$ is isometric to the unit sphere; D. E. Blair [Tôhoku Math. J. 29, 319–324 (1977; Zbl 0376.53021)] studied contact metric manifolds satisfying $$R(X,Y)\xi=0$$, where $$\xi$$ is the characteristic vector field of the contact structure. Contact metric manifolds satisfying $$R(\xi,X)\cdot R=0$$ or $$R(\xi,X)\cdot C=0$$ were studied in [D. Perrone, Yokohama Math. J. 39, 141–149 (1992; Zbl 0777.53046)] and [Ch. Baikoussis and Th. Koufogiorgos, J. Geom. 46, 1–9 (1993; Zbl 0780.53036)], respectively.
The paper under review studies contact metric manifolds satisfying $$Z(\xi,X)\cdot Z=0$$, $$Z(\xi,X)\cdot R=0$$, or $$R(\xi,X)\cdot Z=0$$, where $$Z=R-\frac{r}{n(n-1)}R_{0}$$ with $$R_{0}(X,Y)W=g(Y,W)X-g(X,W)Y$$ is the concircular curvature tensor of $$(M,g)$$. The main results include characterizations which amount to saying that a Riemannian manifold $$(M, g)$$ is a contact metric manifold satisfying $$Z(\xi, X)\cdot Z=0$$ (or $$Z(\xi, X)\cdot R=0$$) and $$R(X,Y)\xi=\kappa R_{0}(X, Y)$$ if and only if $$M$$ is locally isometric to a unit sphere $$S^{2n+1}$$, $$M$$ is locally isometric to an $$N(1-\frac{1}{n})$$-contact metric manifold obtained from $$\mathcal{D}$$-homothetic deformation of the tangent sphere bundle of an $$(n+1)$$-dimensional manifold of constant curvature $$c$$, or $$(M,g)$$ is $$3$$-dimensional and flat.
Reviewer: Ye-Lin Ou (Norman)

### MSC:

 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53D10 Contact manifolds (general theory)

### Citations:

Zbl 0196.25501; Zbl 0376.53021; Zbl 0777.53046; Zbl 0780.53036
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