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On weakly and pseudo-symmetric Riemannian spaces. (English) Zbl 1028.53056
A non-flat Riemannian space $V_n$ ($n>2$) is called a weakly symmetric space, if $$R_{hijk,l}=a_l R_{hijk}+b_h R_{lijk}+c_i R_{hljk}+d_j R_{hilk}+e_k R_{hijl},$$ where $a, b, c, d, e$ are 1-forms (non-zero simultaneously) [{\it L. Tamassy} and {\it T. Q. Binh}, Differential geometry and its applications, Prog. Eger 1989, Colloq. Math. Soc. J. Bolyai 56, 663-670 (1992; Zbl 0791.53021)]. If here $b=c=d=e=\lambda$, $a=2\lambda$, then $V_n$ is called pseudo symmetric [{\it M. C. Chaki}, An. Şţiint. Univ. Al. I. Cuza Iaşi, N. Ser., Secţ. Ia 33, No. 1, 53-58 (1987; Zbl 0626.53037)]. An example is constructed: the metric in $\bbfR^n$ ($n>3$) given by $ds^2=\phi (dx^1)^2+K_{\alpha \beta}dx^\alpha dx^\beta+2dx^1 dx^n$, where $\alpha, \beta$ run over $\{2,3,\dots ,n-1\}$, the matrix of $K_{\alpha \beta}$ is symmetric and non-singular of constants, and $\phi$ is a function of $(x^1, x^2,\dots ,x^{n-1})$ [see {\it W. Roter}, Colloq. Math. 31, 87-96, 97-105 (1974; Zbl 0292.53014, Zbl 0295.53014)]. It is proved that if a totally umbilic hypersurface 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 umbilic hypersurface of a pseudo symmetric space to be pseudo symmetric is obtained. Similar results are obtained for pseudo Ricci symmetric spaces. Also some properties of the Chebyshev and geodesic nets in the hypersurface of these spaces are found.

53C35Symmetric spaces (differential geometry)
53C42Immersions (differential geometry)