Some examples of Riemannian almost-product manifolds
type: Item
instance of: scholarly article
title: Some examples of Riemannian almost-product manifolds
zbMATH Open document ID: 0538.53046
published in: Pacific Journal of Mathematics
publication date: 1984-01-01T00:00:00Z
review text: In the classification of A. M. Naveira for Riemannian almost-product manifolds [Rend. Mat. Appl., VII. Ser. 3, 577--592 (1983; Zbl 0538.53045) (see the previous review)], many of the classes correspond to foliated manifolds with special properties of its metric, such as bundle-like metrics, minimal foliations, totally geodesic foliations, etc. In the present paper, the author gives examples of the 36 classes, showing that these are not vacuous and also that the ten inclusion relations among them are strict. Nine examples are constructed explicitly, and the others are derived from them by means of products and conformal transformations. The basic examples are taken from the following manifolds: a) oriented real hypersurfaces \(M\) in \(\mathbb{C}^{n+1}\). If \(N\) is a unitary normal vector field on \(M\), and \(J\) the canonical complex structure on \(\mathbb{C}^{n+1}\), one defines the distribution on \(M\) generated by the vector field \(JN\). This distribution and its orthogonal define a Riemannian almost-product structure on \(M\); b) oriented real hypersurfaces \(M\) in \(\mathbb{H}^{n+1}\) \((\mathbb{H}= \) quaternion field). If \(J_1, J_2, J_3\) are the canonical almost-complex structures on \(\mathbb{H}^{n+1}\) such that \(J_1J_2 = J_3\), then a Riemannian almost-product structure on \(M\) is defined by means of the distribution generated by \(J_1N\), \(J_2N\), \(J_3N\), and its orthogonal; c) complex hypersurfaces \(M\) in \(\mathbb{H}^{n+1}\) with trivial normal vector bundle. Let \(N\), \(J_1N\) be a frame in the normal vector bundle of \(M\). Then one considers the distribution generated by \(J_2N\), \(J_3N\), and its orthogonal on \(M\).
reviewed by: P. M. Gadea
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C12
zbMATH DE Number: 3855928
zbMATH Keywords: almost-product manifolds
zbMATH Keywords: foliated manifolds
zbMATH Keywords: bundle-like metrics
zbMATH Keywords: minimal foliations
zbMATH Keywords: totally geodesic foliations
Wikidata QID: Q115230777
author: Vicente Miquel
MaRDI profile type: Publication
full work available at URL: pjm.1984.111.163
OpenAlex ID: W1510813922
DOI: 10.2140/PJM.1984.111.163
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description: scientific article; zbMATH DE number 3855928