On some classes of almost contact metric manifolds
type: Item
instance of: scholarly article
title: On some classes of almost contact metric manifolds
zbMATH Open document ID: 0835.53054
author: Jong Taek Cho
published in: Tsukuba Journal of Mathematics
publication date: 1996-03-31T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.tkbjm/1496162808
review text: 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 \(\gamma\).
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 828347
zbMATH Keywords: \(\mathcal C\)-spaces
zbMATH Keywords: tangent sphere bundle of surfaces
zbMATH Keywords: almost contact metric manifold
zbMATH Keywords: Riemannian Jacobi operator
zbMATH Keywords: \(\mathcal P\)-spaces
zbMATH Keywords: real hypersurfaces in complex projective spaces
zbMATH Keywords: Tanaka connection
reviewed by: Jürgen Berndt
MaRDI profile type: Publication
OpenAlex ID: W1916233162
DOI: 10.21099/TKBJM/1496162808
description: scientific article; zbMATH DE number 828347