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On weakly pseudo quasi-conformally symmetric manifolds. (English) Zbl 1167.53023
A non-flat Riemannian manifold ($$M^n,g)$$, $$n>2$$, is called a weakly symmetric manifold if its curvature tensor $$R$$ satisfies the condition
$\begin{split} (\nabla _XR)(Y,Z,U,V)=A(X)R(Y,Z,U,V)+B(Y)R(X,Z,U,V)+\\ +H(Z)R(Y,X,U,V)+D(U)R(Y,Z,X,V)+E(V)R(Y,Z,U,V), \end{split}$ where $$A, B, H, D, E$$ are 1-forms non-zero simultaneously. Such a manifold is denoted by $$(WS)_n.$$
The aim of this paper is to study non-flat Riemannian manifolds ($$M^n,g)$$, $$n>3$$, whose quasi-conformal curvature tensor $$W$$ is not identically zero and satisfies a similar condition. Such a manifold is called a weakly pseudo quasi-conformally symmetric manifold, denoted $$(WPQCS)_n.$$
The quasi-conformal curvature tensor curvature tensor [A. A. Shaikh and S. K. Jana, South East Asian J. Math. Math. Sci. 4, No. 1, 15–20 (2005; Zbl 1116.53016)] is defined by
$\begin{split} W(X,Y,Z,U)=(p+d)R(X,Y,Z,U)+\big(q-\tfrac{d}{ n-1}\big) [S(Y,Z)g(X,U)- S(X,Z)g(Y,U)]+\\ +q[S(X,U)g(Y,Z)-S(Y,U)g(X,Z)]- \tfrac{r}{n(n-1)} [p+2(n-1)q][g(Y,Z)g(X,U)-g(X,Z)g(Y,U)],\end{split}$ where $$R$$ is the curvature tensor of type (0,4), $$S$$ is the Ricci tensor, $$r$$ is the scalar curvature and $$p, q, d$$ are arbitrary constants non-zero simultaneously. This notion includes the conformal, concircular and projective curvature tensor as special cases.
In the present paper the nature of the scalar curvature of $$(WPQCS)_n$$ is studied. Every $$(WS)_n$$ is a $$(WPQCS)_n$$ but not conversely. A sufficient condition for a $$(WPQCS)_n$$ to be $$(WS)_n$$ is obtained. Moreover a sufficient condition for an Einstein $$(WPQCS)_n$$ to be $$(WS)_n$$ is determined. Finally the paper deals with several non-trivial examples of $$(WPQCS)_n$$.

##### MSC:
 53B20 Local Riemannian geometry 53B05 Linear and affine connections 53B35 Local differential geometry of Hermitian and Kählerian structures