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On the existence of a new class of contact metric manifolds. (English) Zbl 0978.53086

Let \(M^{2m+1}(\varphi,\xi,\eta,g)\) be a \((2m+1)\)-dimensional contact metric manifold [cf. e.g. D. E. Blair, Contact manifolds in Riemannian geometry, Lect. Notes Math. 509, Springer-Verlag, Berlin (1976; Zbl 0319.53026)]. Assume that the curvature tensor of the manifold satisfies the condition \[ R(x,y)\xi=\kappa(\eta(y)x-\eta(x)y)+\mu(\eta(y)hx-\eta(x)hy) \] for any vectors \(x\), \(y\) tangent to \(M^{2m+1}\), where \(\kappa\), \(\mu\) are certain functions on \(M^{2m+1}\) and \(h\) is a half of the Lie derivative \({\mathcal L}_{\xi}\varphi\). The main result states that if the manifold is non Sasakian and \(m>1\), then the functions \(\kappa\), \(\mu\) must be constant. Examples of the structures realizing the above condition can be found in the paper by D. E. Blair, T. Koufogiorgos and B. Papantoniou [Isr. J. Math. 91, 189-214 (1995; Zbl 0837.53038)]. In the reviewed paper, examples of 3-dimensional contact metric structures with non constant functions \(\kappa\), \(\mu\) are given.

MSC:

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