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On decomposable weakly conformally symmetric manifolds. (English) Zbl 1224.53057
Let $$(M,g)$$ be an $$n$$-dimensional non-flat Riemannian manifold. A tensor $$C$$ of type $$(0,4)$$ is called the conformal curvature tensor if $\begin{split} C(Y,Z,U,V)=R(Y,Z,U,V)-\frac1{n-2}\left[S(Z,U)g(Y,V)\right.\\ \left.-S(Y,U)g(Z,V)+g(Z,U)S(Y,V)-g(Y,U)S(Z,V)\right]\\ -\frac{r}{(n-1)(n-2)}\left[g(Z,U)g(Y,V)-g(Y,U)g(Z,V)\right], \end{split}$ where $$R$$ and $$S$$ are the curvature and Ricci tensors, respectively, and $$r$$ is the scalar curvature on $$M$$. An $$n$$-dimensional non-flat Riemannian manifold $$(M,g)$$ is called a weakly conformally symmetric manifold if its conformal curvature tensor $$C$$ of type $$(0, 4)$$ satisfies $\begin{split}(\nabla_ X C)(Y,Z,U,V)= \alpha(X) C(Y,Z,U,V)\\ +\beta(Y) C(X,Z,U,V)+\gamma(Z) C(Y,X,U,V)\\ +\delta(U)C(Y,Z,X,V)+\sigma(V) C(Y,Z,U,X)), \end{split}$ where $$\alpha$$, $$\beta$$, $$\gamma$$, $$\delta$$, and $$\sigma$$ are $$1$$-forms on $$M$$. It turns out that $$\beta=\gamma$$ and $$\delta=\sigma$$. A Riemannian manifold $$(M,g)$$ is said to be decomposable if it can be expressed as the product $$M_ 1^ p\times M_ 2^{n-p}$$.
In this paper, the authors study properties of weakly conformally symmetric manifolds. It is shown that if a weakly conformally symmetric manifold $$(M,g)$$ is decomposable with $$M=M_ 1\times M_ 2$$, then either $$M_ 1$$ is locally symmetric or $$M_ 2$$ is of constant curvature. Some examples of weakly conformally symmetric manifolds are presented.

##### MSC:
 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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