Abstract
It is proved that every concircularly recurrent manifold must be necessarily a recurrent manifold with the same recurrence form.
Similar content being viewed by others
References
E. Boeckx, O. Kowalski and L. Vanhecke, Riemannian Manifolds of Conullity Two, World Scientific (Singapore–New Jersey–London–Hong Kong, 1996).
K. Arslan, U. C. De, C. Murathan and A. Yildiz, On generalized recurrent Riemannian manifolds, Acta Math. Hungar., 123 (2009), 27–39.
D. E. Blair, J.-S. Kim and M. M. Tripathi, On the concircular curvature tensor of a contact metric manifold, J. Korean Math. Soc., 42 (2005), 883–892.
U. C. De and K. Gazi, On generalized concircularly recurrent manifolds, Studia Math. Hung., 46 (2009), 287–296.
U. C. De and N. Guha, On generalized recurrent manifolds, J. Nat. Acad. Math. India, 9 (1991), 85–92.
R. S. D. Dubey, Generalized recurrent spaces, Indian J. Pure Appl. Math., 10 (1979), 1508–1513.
V. F. Kirichenko and E. A. Pol’kina, A criterion for the concircular mobility of quasi-Sasakian manifolds, Math. Notes, 86 (2009), 349–356; translated from Mat. Zametki, 86 (2009), 380–388.
V. F. Kirichenko and L. I. Vlasova, Concircular geometry of nearly Kählerian manifolds, Sb. Math., 193 (2002), 685–707; translation from Mat. Sb., 193 (2002), 53–76.
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. I, Interscience Publishers, a division of John Wiley & Sons (New York–London, 1963).
C. A. Mantica and L. G. Molinari, A second-order identity for the Riemann tensor and applications, Colloq. Math., 122 (2011), 69–82.
Y. B. Maralabhavi and M. Rathnamma, Generalized recurrent and concircular recurrent manifolds, Indian J. Pure Appl. Math., 30 (1999), 1167–1171.
P. J. Ryan, A class of complex hypersurfaces, Colloq. Math., 26 (1972), 175–182.
H. S. Ruse, A. G. Walker and T. J. Willmore, Harmonic Spaces, Ed. Cremonese (Roma, 1961).
H. Singh and Q. Khan, On generalized recurrent Riemannian manifolds, Publ. Math. Debrecen, 56 (2000), 87–95.
L. Vanhecke, Curvature tensors, J. Korean Math. Soc., 14 (1977), 143–151.
A. G. Walker, On Ruses’s spaces of recurrent curvature, Proc. London Math. Soc., 52 (1950), 36–64.
Y.-C. Wong, Recurrent tensors on a linearly connected differentiable manidfold, Trans. Amer. Math. Soc., 99 (1961), 325–341.
Y.-C. Wong, Linear connexions with zero torsion and recurrent curvature, Trans. Amer. Math. Soc., 102 (1962), 471–506.
K. Yano, Concircular geometry. I. Concircular transformations, II. Integrability conditions of ϱ μν =Φg μν , III. Theory of curves, IV. Theory of subspaces, V. Einstein spaces, Proc. Imp. Acad. Tokyo, 16 (1940), 195–200, 354–360, 442–448, 505–511, 18 (1942), 446–451.
Author information
Authors and Affiliations
Corresponding author
Additional information
Corresponding author.
Rights and permissions
About this article
Cite this article
Olszak, K., Olszak, Z. On pseudo-Riemannian manifolds with recurrent concircular curvature tensor. Acta Math Hung 137, 64–71 (2012). https://doi.org/10.1007/s10474-012-0216-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-012-0216-5