On pseudo symmetric spaces
A non-flat Riemannian space whose curvature tensor \(R_{hijk}\) satisfies the condition \[ R_{hijk,\ell}=2\lambda_{\ell}R_{hijk}+\lambda_ hR_{\ell ijk}+\lambda_ iR_{h\ell jk}+\lambda_ jR_{hi\ell k}+\lambda_ kR_{hij\ell}, \] where \(\lambda_ i\) is a non-zero vector and comma denotes covariant differentiation with respect to the metric tensor of the space, is called a pseudo symmetric space, denoted by \((PS)_ n\) and \(\lambda_ i\) is called its associated vector field [the first author, An. Stiint. Univ. Al I. Cuza Iasi, N. Ser., Sect. Ia 33, No.1, 53-58 (1987; Zbl 0626.53037)]. In this paper, the authors study various characterizations of such a pseudo symmetric Riemannian space and discuss the properties of this space and of several vector fields, e.g. the Ricci associate of \(\lambda_ i\), concurrent and recurrent vector fields admitted by \((PS)_ n\), etc.
scientific article; zbMATH DE number 4131181