## On generalized Ricci-recurrent manifolds.(English)Zbl 0859.53009

The authors propose a new generalization of the notion of a Ricci-recurrent manifold $$R_n$$ introduced in 1952 by E. M. Patterson. Their generalized Ricci-recurrent manifold $$GR_n$$ is a nonflat Riemannian manifold which possesses a pair of nonzero 1-forms: a recurrence 1-form $$A$$ and an associated 1-form $$B$$. When $$B=0$$, then $$GR_n$$ reduces to a $$R_n$$. A different notion of a generalized Ricci-recurrent manifold $$GK_n$$ was proposed by the first two authors in 1991 (apparently not yet published). It reduces to the original H. S. Ruse and A. G. Walker definition of recurrent manifolds $$K_n$$ when $$B=0$$.
The goal of the current investigation is to study the new notion of a generalized Ricci-recurrent manifold and ascertain when a $$GR_n$$ reduces to a $$GK_n$$. Ultimately the issue is to determine whether the appropriate generalization of Ricci-recurrent manifolds should be closer to the Ricci-recurrent manifolds of Patterson, or the recurrent manifolds of Ruse and Walker. Contents include: an introduction; preliminaries; existence of a $$GR_n$$ $$(n \geq 2)$$; the 1-forms $$A$$ and $$B$$; $$GR_n$$ with constant scalar curvature; conformally flat $$GR_n$$ with constant scalar curvature; and a necessary and sufficient condition for a $$GR_n$$ to be a $$GK_n$$.

### MSC:

 53B20 Local Riemannian geometry