On recurrent spaces with semi-symmetric Singh connection. (Q1411774)

Let \(V_{n}\) be an \(n\)-dimensional Riemannian space \((n>2)\) endowed with a symmetric tensor \(g_{ij}\) such that \(det (g_{ij}) \neq 0\). Denote by \(\gamma_{ij}^{k}\) the Christoffel symbols of \(g_{ij}\). A connection \(\Gamma_{ij}^{h}\) of \(V_{n}\) is said to be a Singh connection if it satisfies the following equation \[ \partial_{h} g_{ij} -g_{kj} \Gamma_{hi}^{k} -g_{ik} \Gamma_{hj}^{k} = 0. \] In particular a semi-symmetric Singh connection of \(V_{n}\) is given by \[ \Gamma_{ij}^{h} = \gamma_{ij}^{h} -2g_{ij} X^{h} + 2 \delta_{i}^{h} X_{j}, \] for some vector \(X_{j}\), where \(X^{h} := g^{jk} X_{j}\). The curvature tensor \(L_{ijk}^{h}\) of \(V_{n}\) endowed with a semi-symmetric Singh connection is given by \[ L_{ijk}^{h} = R_{ijk}^{h} -2 \delta_{i}^{h} (P_{jk}-P_{kj}) +g_{ij} P_{k}^{h}-g_{ik}P_{j}^{h}, \] where \(R_{ijk}^{h}\) is the Riemannian curvature tensor of the space \(V_{n}\) and \[ P_{jk} = \nabla_{j} X_{k} - 2 X_{j} X_{k}, \, P_{j}^{h} = g^{kh} P_{jk}. \] Here \(\nabla_{j}\) denotes the symbol of the covariant derivative with respect to the Riemannian connection. Introduce the tensor \[ L_{hijk} := R_{hijk} -2g_{ih} (P_{jk} - P_{kj}) +g_{ij} P_{kh} -g_{ik} P_{jh}. \] A non-flat space \(V_{n}\) endowed with a semi-symmetric Singh connection is called recurrent if the curvature tensor field of the space satisfies the following condition \[ \nabla_{l} L_{hijk} = \lambda_{l} L_{hijk}, \] for some non-zero covariant vector field \(\lambda_{l}\). In the paper under review the authors obtain an example of a recurrent space with a semi-symmetric Singh connection.