On the submanifolds of an almost para-Hermitian manifold
An almost para-Hermitian manifold \((\overline M,\overline J, \overline g)\) is a differentiable manifold \(\overline M\) endowed with an almost product structure \(\overline J\), that is, a tensor field \(\overline J\) of type \((1,1)\) obeying \(\overline J\circ\overline J=Id\), and a pseudo-Riemannian metric \(\overline g\) such that \(\overline g(\overline JX,Y)+\overline g(X,\overline JY)=0\), for all vector fields \(X\) and \(Y\). In this paper, the authors give the classification of degenerate and non-degenerate submanifolds, and CR-submanifolds, of an almost para-Hermitian manifold of dimension four. They provide also examples from each class. As a consequence of that classification, the authors find that there exist submanifolds which admit more than one CR-structure, revealing a profound difference between complex and para-complex geometries.
scientific article; zbMATH DE number 1528628