On generalized Ricci 3-recurrent spaces (Q2771181)

Let \((M^n,g)\) be a non-flat \(n\)-dimensional Riemannian manifold with \(n\geq 3\) and let \(\nabla\) be its Levi Civita connection and \(R\) the Riemannian curvature tensor of type \((0,4)\). \((M,g)\) is said to be a generalized 3-recurrent space (denoted by \(G(3k_n))\) if the condition NEWLINE\[NEWLINE\nabla^3R= \lambda\otimes \nabla^2R+ \mu \otimes R\tag{*}NEWLINE\]NEWLINE is satisfied for a non-zero one-form \(\lambda\) and a non-zero \((0,3)\)-tensor \(\mu\). Such spaces have been introduced and studied before and examples have been provided. The condition (*) implies NEWLINE\[NEWLINE\nabla^3\rho= \lambda\otimes \nabla 62\rho +\mu\otimes \rho \tag{**}NEWLINE\]NEWLINE where \(\rho\) denotes the Ricci tensor of type \((0,2)\). The main purpose of the authors is to consider \((M^n,g)\) which satisfy (**) and which are not Ricci-flat. They are denoted by \(G(2R_n)\) in the paper. The above remark shows that a non-Ricci flat \(G(3k_n)\) is a \(G(3R_n)\) and the authors focus on the problem whether the converse is true. First, they prove that this is indeed the case for \(n=3\) and more generally, for any \((M^n,g)\) which is conformally symmetric, i.e. the Weyl tensor \(C\) is parallel. Finally, they show, by giving an example, that the converse does not hold in general.