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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 [TĂ´hoku 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)
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