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 \(GK_n\).


53B20 Local Riemannian geometry