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Zbl 1028.53056
Özen, Füsun; Altay, Sezgin
On weakly and pseudo-symmetric Riemannian spaces.
(English)
[J] Indian J. Pure Appl. Math. 33, No.10, 1477-1488 (2002). ISSN 0019-5588; ISSN 0975-7465/e

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.
[Ülo Lumiste (Tartu)]
MSC 2000:
*53C35 Symmetric spaces (differential geometry)
53C42 Immersions (differential geometry)

Keywords: weakly symmetric Riemannian space; pseudo-symmetric Riemannian space; hypersurface

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