On some classes of contact metric manifolds. (English) Zbl 0817.53019

Cho, Jong Taek

Rend. Circ. Mat. Palermo, II. Ser. 43, No. 1, 141-160 (1994).

Author’s abstract: “We define four new classes \((C_{\overline {\nabla}} ,p_{\overline \nabla}, DC_{\overline{\nabla}}, D_{p \overline{\nabla}})\) of contact metric manifolds using the Tanaka connection [N. Tanaka, Jap. J. Math., New Ser. 2, 131-190 (1976; Zbl 0346.32010)] and Jacobi operators. We prove that a contact metric manifold with the structure vector field \(\zeta\) belonging to the \(K\)- nullity distribution is contact metric locally \(\phi\)-symmetric [D. E. Blair, Contact manifolds in Riemannian geometry, Lect. Notes Math. 509 (1976; Zbl 0319.53026)] if and only if the manifold is a \(DC_{\overline{\nabla}}\) and \(D_{p_{\overline\nabla}}\)-space. Also we prove that a 3-dimensional contact metric \(C_{\overline{\nabla}}\) and \(p_{\overline{\nabla}}\)-space is locally \(\phi\)-symmetric and give counter-examples of the converse”.

Reviewer: R.Roşca (Paris)

MSC:

53C15

General geometric structures on manifolds (almost complex, almost product structures, etc.)

Keywords:

\(\phi\)-symmetric space; tangent sphere bundle; Tanaka connection; Jacobi operators

Citations:

Zbl 0346.32010; Zbl 0319.53026

References:

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