Generalized \((\kappa, \mu)\)-contact metric manifolds with \(\| \operatorname{grad} \kappa\| = \) constant
The generalized \((\kappa, \mu)\)-contact metric manifolds (briefly, generalized \((\kappa, \mu)\)-c.m.m.) were introduced by the authors in [Can. Math. Bull. 43, 440--447 (2000; Zbl 0978.53086)]. Such a manifold is a contact metric manifold \((\eta, \xi, \phi, g)\) that satisfies a specific identity for the curvature tensor \(R\), namely: \(R(X, Y) \xi = (\kappa I+\mu h)(\eta(Y)X-\eta(X)Y)\) (here, \(h\) is the Lie derivative of \(\phi\) in the direction \(\xi\)). If \(\dim M > 3\), then \(\kappa\) and \(\mu\) are necessarily constant. Consequently, the most important examples of such varieties have necessarily dimension 3. In this case D. Perrone gives a geometric interpretation in terms of harmonic maps. The existence of such a class of generalized \((\kappa, \mu)\)-c.m.m. is proved by two examples. Moreover, is proved that for every positive number, there exist at least two generalized \((\kappa, \mu)\)-c.m.m. with \(\|\text{grad\,} \kappa \|= \text{constant} \neq 0\). Every such generalized \((\kappa, \mu)\)-c.m.m. is a parallelizable manifold. In this paper, the authors give an analytic local classification of the generalized \((\kappa, \mu)\)-c.m.m. with the additional condition \(\|\text{grad\,} \kappa \|= \text{constant} \neq 0\). This class depends on two arbitrary functions of one variable.
scientific article; zbMATH DE number 2067298