Hypersurfaces of an almost \(r\)-paracontact Riemannian manifold with a semi-symmetric non-metric connection
For an \(n\)-dimensional manifold \(\widetilde M^n\) with Riemannian metric \(g\) the structure \(\Sigma= (\varphi,\xi_\alpha,\eta^\alpha,\) \(g)_{\alpha\in (\tau< n)}\) is said to be almost \(r\)-paracontact if its members satisfy \[ \eta^\alpha(\xi_\beta)= \delta^\alpha_\beta,\quad \varphi^2(X)= X- \eta^\alpha(X)\xi_\alpha,\quad \eta^\alpha\circ\varphi= 0, \] \[ \eta^\alpha(X)= g(X,\xi_\alpha),\quad g(\xi X,\varphi Y)= g(X,Y)- \sum_\alpha \eta^\alpha(X) \eta^\alpha(Y),\quad (\alpha,\beta\in(r)) \] and \(\widetilde M^n\) as such is called an almost \(r\)-paracontact Riemannian manifold. Suppose now that \(\widetilde M^n\) is endowed with a connection such that it is semi-symmetric, that is, its torsion tensor satisfies \(T(X,Y)= u(Y)X-u(X)Y\), and also is non-metric, that is, it is not the Levi-Cività connection. Let \(M^{n-1}\) be a hypersurface of \(\widetilde M^n\). It is then shown that the connection, induced with respect to the unit normal on \(M^{n-1}\), is also semi-symmetric and non-metric. These hypersurfaces are classified into three. They are invariant, non-invariant and anti-invariant ones. If \(M^{n-1}\) is invariant in \(\widetilde M^n\) and \(\widetilde M^n\) is of para-Sasakian type, then the induced almost \(r\)-paracontact Riemannian structure \(\Sigma_1\) or the \((r-1)\)-paraconact Riemannian structure \(\Sigma_2\) is also of para-Sasakian type.
scientific article; zbMATH DE number 5627196