Some examples of Riemannian almost-product manifolds
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 \(M\).
scientific article; zbMATH DE number 3855928