A note on generalized recurrent Riemannian manifold

Abstract

Generalized recurrent Riemannian manifold is a Riemannian manifold whose curvature tensor satisfies the condition (1). In this paper we prove: If the associated $1$-form satisfies the condition (3), where $\gamma\ne1\ne2$, or, in the case $\gamma\ne\text{const}$, $\gamma_sA^s\ne0$, generalized recurrent Riemannian manifold reduces to a recurrent one.