On some classes of almost contact metric manifolds. (English) Zbl 0835.53054

Cho, Jong Taek

Tsukuba J. Math. 19, No. 1, 201-217 (1995).

It is known that a Riemannian manifold \((M,g)\) is locally symmetric if and only if along any geodesic \(\gamma\) in \(M\) the associated Riemannian Jacobi operator \(R_\gamma := R(.,\dot \gamma) \dot \gamma\) has constant eigenvalues and is diagonalizable by a parallel orthonormal frame field. If in a Riemannian manifold \((M,g)\) all \(R_\gamma\) satisfy just the first or the second of these two properties, then \((M,g)\) is called a \(\mathcal C\)-space or a \(\mathcal P\)-space, respectively.
The author studies \(\mathcal C\)- and \(\mathcal P\)-spaces in the class of almost contact metric manifolds \((M, \varphi, \xi, \eta, g)\) by assuming in addition that \(\nabla_\xi \xi = 0\) and \(\nabla_\xi R_\xi = 0\), where \(\nabla\) is the Levi Civita connection of \((M,g)\). Denote such a manifold by \(\xi {\mathcal C}\)- and \(\xi{\mathcal P}\)-space, respectively. He proves that the tangent sphere bundle \(T_1 M\) of a 2-dimensional Riemannian manifold \((M,g)\), equipped with the almost contact metric structure induced from the natural almost Kähler structure on \(TM\), is a \(\xi {\mathcal C}\)-space if and only if \((M,g)\) has constant curvature. And it is a \(\xi {\mathcal P}\)-space if and only if \((M,g)\) has constant curvature zero or one. He also studies \(\xi {\mathcal C}\)- and \(\xi {\mathcal P}\)-spaces among real hypersurfaces in complex projective spaces. Eventually, the author studies analogous questions for \({\mathcal D} {\mathcal C}\)- and \({\mathcal D} {\mathcal P}\)-spaces. Such spaces are almost contact metric manifolds, and \({\mathcal D} {\mathcal C}\) resp. \({\mathcal D} {\mathcal P}\) means that along any horizontal geodesic \(\gamma\) with respect to the Tanaka connection \(\overline {\nabla}\) the associated Riemannian Jacobi tensor \(R_\gamma\) has constant eigenvalues resp. is diagonalizable by a \(\overline {\nabla}\)-parallel orthonormal frame field along \(\gamma\).

Reviewer: J.Berndt (Köln)

MSC:

53C25	Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C15	General geometric structures on manifolds (almost complex, almost product structures, etc.)

Keywords:

\(\mathcal C\)-spaces; tangent sphere bundle of surfaces; almost contact metric manifold; Riemannian Jacobi operator; \(\mathcal P\)-spaces; real hypersurfaces in complex projective spaces; Tanaka connection