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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.

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