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Zbl 1136.53019
Shaikh, Absos Ali; Jana, Sanjib Kumar
On weakly symmetric Riemannian manifolds.
(English)
[J] Publ. Math. 71, No. 1-2, 27-41 (2007). ISSN 0033-3883

A non-flat Riemannian manifold $(M^n,g)$ $(n>2)$ is called weakly symmetric -- and the denotes by $(WS)_n$ -- if its curvature tensor $R$ of type $(0,4)$ satisfies the condition: \aligned & (\nabla_XR)(Y,Z,U,V)=A(X)\cdot R(Y,Z,U,V)+B(Y)\cdot R(X,Z,U,V)\\ &\qquad +C(Z)\cdot R(Y,X,U,V) + D(U)\cdot R(Y,Z,X,V)+E(V)\cdot R(Y,Z,U,X) \endaligned for all vector fields $X,Y,Z,U,V\in \chi (M^n)$, where $A$, $B$, $C$, $D$ and $E$ are 1-forms (non-zero simultaneously) and $\nabla$ is the operator of covariant differentiation with respect to $g$. The present note on $(WS)_n$ consists of 4 sections starting with Introduction" and Fundamental results of a $(WS)_n$ $(n>2)$". In Section 3 on Conformally flat $(WS)_n$"  the authors show -- among others -- that a conformally flat $(WS)_n$ $(n>3)$ of non-zero scalar curvature is of hyper quasi-constant curvature (which generalizes the notion of quasi-constant curvature) and also such a manifold is a quasi-Einstein manifold (Theorems 6--9). Finally (Section~4), several examples of $(WS)_n$ of both zero and non-zero scalar curvature are obtained, in particular a manifold $(WS)_n$ $(n\ge 4)$ which is neither locally symmetric nor recurrent, the scalar curvature of which is vanishing (Theorem~12) or non-vanishing and non-constant (Theorem~14), respectively.
[Richard Koch (München)]
MSC 2000:
*53B35 Complex differential geometry (local)
53B05 Linear and affine connections

Keywords: weakly symmetric manifold; conformally flat manifold; hyper-quasi constant curvature; scalar curvature

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