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On the existence of certain types of Riemannian metrics. (English) Zbl 0787.53015

An \(n\)-dimensional Riemannian manifold \((M,g)\) (a metric \(g\) need not be definite) is said to be \(s\)-recurrent if its Riemann curvature tensor \(R\) satisfies the condition \[ R_{hijk,l_ 1,\dots,l_ s}R_{ptqr} - R_{hijk}R_{pqtr,l_ 1,\dots,l_ s} = 0, \] where the comma denotes covariant derivative with respect to \(g\). This condition implies that for each \(x\in M\) such that \(R(x)\neq 0\) there exists a unique tensor \(a\) of type \((0,s)\) which satisfies at \(x: R_{hijk,l_ 1,\dots,l_ s} = a_{l_ 1,\dots,l_ s}R_{hijk}\). In the same way conformally \(s\)- recurrent manifolds are defined. A manifold \((M,g)\) will be called \(s\)- symmetric if the tensor \(R\) satisfies the condition \(R_{hijk,l_ 1,\dots,l_ s} = 0\). In the reviewed paper the author investigates some properties of conformally \(p\)-symmetric manifolds with recurrent Ricci tensor and establishes the existence of such manifolds which are neither conformally \(s\)-recurrent, \(s = 1,\dots,p-1\), nor \(p\)-recurrent. Moreover, he proves that for each \(n>4\) and \(p\geq 2\) there exists an essentially conformally \(p\)-recurrent manifold with recurrent Ricci tensor.

MSC:

53B20 Local Riemannian geometry
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