Bejancu, Aurel Kähler contact distributions. (English) Zbl 1209.53031 J. Geom. Phys. 60, No. 12, 1958-1967 (2010). 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. Reviewer: D. Perrone (Lecce) Cited in 12 Documents MSC: 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) Keywords:contact metric manifold; \(\mathcal D\)-connection; Kähler contact distribution; Sasakian manifold; Sasakian space form PDFBibTeX XMLCite \textit{A. Bejancu}, J. Geom. Phys. 60, No. 12, 1958--1967 (2010; Zbl 1209.53031) Full Text: DOI References: [1] Blair, D. E., Riemannian Geometry of Contact and Symplectic Manifolds (2001), Birkhäuser: Birkhäuser Basel [2] Tanno, S., Variational problems on contact Riemannian manifolds, Trans. Amer. Math. Soc., 314, 349-379 (1989) · Zbl 0677.53043 [3] Ogiue, K., On almost contact manifolds admitting axiom of planes or axiom of free mobility, Kōdai Math. Sem. Rep., 16, 223-232 (1964) · Zbl 0136.18003 [4] O’Neill, B., Semi-Riemannian Geometry with Applications to Relativity (1983), Academic Press: Academic Press New York · Zbl 0531.53051 [5] Yano, K.; Kon, M., Structures on Manifolds (1984), World Scientific: World Scientific Singapore · Zbl 0557.53001 [6] Kobayashi, S.; Nomizu, K., Foundations of Differential Geometry, vol. II (1969), Interscience: Interscience New York · Zbl 0175.48504 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.