Contact metric manifolds satisfying a nullity condition
The authors study contact metric manifolds \(M^{2n+ 1}(\varphi, \xi, \eta, g)\) [for definitions see \textit{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.
scientific article; zbMATH DE number 820098