On weakly and pseudo-symmetric Riemannian spaces

It is proved that if the totally umbilical hypersurface (M ≠ 0) of a weakly symmetric space is a weakly symmetric space then it is a pseudo symmetric space. A necessary and sufficient condition for a totally umbilical hypersurface of a pseudo symmetric space to be a pseudo symmetric is obtained. In addition, it is shown that if the totally umbilical hypersurface of a pseudo Ricci symmetric space is pseudo Ricci symmetric then this space is of zero scalar curvature and the condition M_,h- λ_h M = 0 is satisfied. Finally, we study some properties of the Chebyshev and geodesic nets in the hypersurface of these spaces.