Kähler contact distributions
Let \(M(\varphi ,\xi ,\eta ,g)\) be a contact metric manifold and \(\mathcal D\) be its contact distribution. We define a linear connection \(\nabla \) on \(M\) and relate it with the generalized Tanaka connection introduced by Tanno in 1989. This enables us to give a characterization in terms of \(\nabla \) of a strongly pseudo-convex \(CR\)-structure on \(M\). Then we prove that \(\nabla \varphi =0\) if and only if \(M\) is a Sasakian manifold. In this case we call \((\mathcal D, \varphi , g)\) a Kähler contact distribution. Finally, we prove that such a \((\mathcal D, \varphi , g)\) has constant holomorphic sectional curvature with respect to \(\nabla \) if and only \(M\) is a Sasakian space form.
scientific article; zbMATH DE number 5819533