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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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References:
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