On quasi-conformally flat weakly Ricci symmetric manifolds.(English)Zbl 1164.53011

The authors obtain certain properties of (pseudo-)Riemannian manifolds which are weakly Ricci symmetric and simultaneously quasi-conformally flat. The reader must be careful because the formulations of the theorems are not complete. For instance, Theorem 1 states that the Ricci tensor is of rank 1. In this theorem, it is in fact additionally assumed that $$\delta\neq0$$, which eliminates a large subclass of the considered manifolds. Moreover, the examples are not well described. For instance, the metric in Example 1 is Ricci recurrent, which contradicts Theorem 11, where the non-Ricci recurrence of this metric is asserted.

MSC:

 53B20 Local Riemannian geometry 53B30 Local differential geometry of Lorentz metrics, indefinite metrics 53B35 Local differential geometry of Hermitian and Kählerian structures 53B05 Linear and affine connections
Full Text:

References:

 [1] T. Adati, On subprojective spaces III, Tohoku Math. J., 3 (1951), 343–358. · Zbl 0045.11202 [2] K. Amur and Y. B. Maralabhavi, On quasi-conformally flat spaces, Tensor N. S., 31 (1977), 194–198. · Zbl 0362.53010 [3] T. Q. Binh, On weakly symmetric Riemannian spaces, Publ. Math. Debrecen, 42 (1993), 103–107. · Zbl 0797.53041 [4] M. C. Chaki and S. Koley, On generalized pseudo Ricci symmetric manifolds, Periodica Math. Hungar., 28 (1993), 123–129. · Zbl 0821.53017 [5] M. C. Chaki and R. K. Maity, On quasi-Einstein manifolds, Publ. Math. Debrecen, 57 (2000), 297–306. · Zbl 0968.53030 [6] B. Y. Chen and K. Yano, Hypersurfaces of a conformally flat space, Tensor N. S., 26 (1972), 318–322. · Zbl 0257.53027 [7] B. Y. Chen and K. Yano, Special conformally flat spaces and Canal hypersurfaces, Tohoku Math. J., 25 (1973), 177–184. · Zbl 0266.53043 [8] J. A. Schouten, Ricci-Calculus, Springer-Verlag (Berlin, 1954). [9] L. Tamássy and T. Q. Binh, On weakly symmetric and weakly projective symmetric Riemannian manifolds, Coll. Math. Soc. J. Bolyai, 50 (1989), 663–670. · Zbl 0791.53021 [10] L. Tamássy and T. Q. Binh, On weak symmetrics of Einstein and Sasakian manifolds, Tensor N. S., 53 (1993), 140–148. · Zbl 0849.53038 [11] K. Yano, Concircular geometry, I, Proc. Imp. Acad. Tokyo, 16 (1940), 195–200. · Zbl 0024.08102 [12] K. Yano, On the torseforming direction in Riemannian spaces, Proc. Imp. Acad., Tokyo, 20 (1944), 340–345. · Zbl 0060.39102 [13] K. Yano and S. Sawaki, Riemannian manifolds admitting a conformal transformation group, J. Diff. Geom., 2 (1968), 161–184. · Zbl 0167.19802
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.