We introduce a new kind of Riemannian manifold that includes weakly-, pseudo- and pseudo projective Ricci symmetric manifolds. The manifold is defined through a generalization of the so called Z tensor; it is named weakly Z-symmetric and is denoted by (WZS) n . If the Z tensor is singular we give conditions for the existence of a proper concircular vector. For non singular Z tensors, we study the closedness property of the associated covectors and give sufficient conditions for the existence of a proper concircular vector in the conformally harmonic case, and the general form of the Ricci tensor. For conformally flat (WZS) n manifolds, we derive the local form of the metric tensor.
Key words and phrasesweakly-Ricci symmetric manifold pseudo- projective Ricci symmetric conformal curvature tensor quasi conformal curvature tensor conformally symmetric conformally recurrent Riemannian manifolds weakly Z-symmetric manifold
2010 Mathematics Subject Classification53B20 53B21
Unable to display preview. Download preview PDF.
- F. Defever and R. Deszcz, On semi Riemannian manifolds satisfying the condition R⋅R=Q(S,R), in: Geometry and Topology of Submanifolds, III, Leeds, May 1990, World Sci. (Singapore, 1991), pp. 108–130. Google Scholar
- L. P. Eisenhart, Non Riemaniann Geometry, reprint Dover Ed. (2005). Google Scholar
- A. Gebarowsky, Nearly conformally symmetric warped product manifolds, Bull. Inst. Math., Acad. Sin., 20 (1992), 359–371. Google Scholar
- D. Lovelock and H. Rund, Tensors, differential forms and variational principles, reprint Dover Ed. (1988). Google Scholar
- C. A. Mantica and L. G. Molinari, Extended Derdzinski–Shen theorem for the Riemann tensor, arXiv:1101.4157 [math.DG], 21 Jan 2011.
- J. A. Shouten, Ricci-Calculus, 2nd ed. Springer Verlag (1954). Google Scholar