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.


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


Zbl 0319.53026
Full Text: DOI


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