On generalized pseudo symmetric manifolds. (English) Zbl 0889.53015

Chaki, M. C.; Mondal, S. P.

Publ. Math. 51, No. 1-2, 35-42 (1997).

A Riemannian space \(V^n\) is locally symmetric if \(\nabla_XR=0\), and of recurrent curvature if \((*)\) \(\nabla_XR=a(X)R\) for some 1-form \(a\). This notion was weakened by M. C. Chaki [Publ. Math. 45, 305-312 (1994; Zbl 0827.53032)] and L. Tamássy and T. Q. Binh [Colloq. Math. Soc. János Bolyai 56, 663-670 (1992; Zbl 0791.53021)] by replacing \((*)\) with \((\nabla_XR)(Y,V,W,Z)= \sigma(X,Y,V,W,Z)\) \(\{\overset\sigma a(X)R(Y,V,W,Z)\}\) (\(\sigma\) means the cyclic summation). Such a \(V^n\) is called a generalized pseudosymmetric space \(G(PS)_m\). Now it is proved that some of the 1-forms \(\overset\sigma a\) of a \(G(PS)_n\) necessarily coincide. Conformally flat \(G(PS)_n\) with constant scalar curvature are also investigated.

Reviewer: Lajos Tamássy (Debrecen)

MSC:

53B20

Local Riemannian geometry

Keywords:

pseudo-symmetric manifolds; conformal flatness; constant scalar curvature