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On weakly symmetric Riemannian manifolds. (English) Zbl 0860.53010

A nonflat Riemannian manifold \(V^n\) is called weakly symmetric (w.s.) if \(\nabla_r R_{hijk}=A_r R_{hijk}+B_h R_{rijk}+C_i R_{hrjk}+D_jR_{hirk}+E_k R_{hijr}\) [see T. Q. Binh and L. Tamássy, Colloq. Math. Soc. János Bolyai 56, 663-670 (1992; Zbl 0791.53021), T. Q. Binh, Publ. Math. 42, No. 1-2, 103-107 (1993; Zbl 0797.53041)]. It is called pseudo-symmetric (p.s.) if \(B=C=D=E={1\over 2}A\) [see M. C. Chaki and U. C. De, Acta Math. Hung. 54, No. 3/4, 185-190 (1989; Zbl 0691.53011)]. \(V^n\) is called a \(\mathcal B\)-space if there exists a nonzero vector field \(\theta\) such that \(\theta_l R_{hijk}+\theta_jR_{hikl} +\theta_k R_{hilj}=0\) [see P. Venzi, Rev. Roum. Math. Pures Appl. 30, 295-305 (1985; Zbl 0584.53015)]. The author proves that a w.s. \(V^n\) is either p.s. or it is a \(\mathcal B\)-space. She gets necessary and sufficient conditions for a space with \(\dim\{\theta\}=1\) to be w.s. and shows that in this case \(B=C=D=E\).

MSC:

53B20 Local Riemannian geometry
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