On the geometry of paracomplex submanifolds (Q2777566)

The paracomplex submanifolds \(M\) in an almost parahermitian manifold (\(\overline M,\overline J,\overline g)\) with \(\overline J^2=I\), \(\overline g(X,\overline JY)+\overline g(\overline JX,Y)=0\) are characterized by \(\overline J(T_xM)=T_xM\) [see \textit{V. Cruceanu}, \textit{P. Fortuny} and \textit{P. M. Gradea} in Rocky Mt. J. Math. 26, 83-115 (1996; Zbl 0856.53049)]. In the present paper the induced tensor field \(g\) induced on \(M\) by \(\overline g\) is supposed to be nondegenerate and of constant index. Then \((M,J,g)\), where \(J\) is induced by \(\overline J\), is an almost parahermitian submanifold; moreover, if \((\overline M,\overline J,\overline g)\) is parakählerian, this \(M\) is parakählerian and a minimal submanifold. (The last result generalizes a result of the reviewer of 1957 (see Zbl 0080.14505), and its further generalizations.) NEWLINENEWLINENEWLINEIn parakählerian space forms of constant curvature \(c\) the submanifolds are characterized, which are parakählerian space forms of the same curvature \(c\), or which are Einstein manifolds. Specially the degenerate (in the sense of \(g)\) parahermitian hypersurfaces \(M\) of parahermitian manifolds are investigated. It is shown that (1) the lightlike distribution on such an \(M\) is invariant with respect to \(J\), (2) there exist no such hypersurfaces \(M\) with lightlike distribution of rank 1. (3) there exists a screen distribution on \(M\) that is invariant with respect to \(J\), (4) there exists a transversal vector bundle \(tr(TM)\) that is invariant with respect to \(\overline J\).NEWLINENEWLINENEWLINEA general result is proved which in lower dimension leads to the consequence that any degenerate parahermitian surface of a 4-dimensional parahermitian manifold is totally geodesic.