## On pseudo-Riemannian manifolds with recurrent concircular curvature tensor.(English)Zbl 1299.53110

Summary: It is proved that every concircularly recurrent manifold must be necessarily a recurrent manifold with the same recurrence form.

### MSC:

 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics 53B20 Local Riemannian geometry 53B30 Local differential geometry of Lorentz metrics, indefinite metrics
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### References:

 [1] E. Boeckx, O. Kowalski and L. Vanhecke, Riemannian Manifolds of Conullity Two, World Scientific (Singapore–New Jersey–London–Hong Kong, 1996). · Zbl 0904.53006 [2] K. Arslan, U. C. De, C. Murathan and A. Yildiz, On generalized recurrent Riemannian manifolds, Acta Math. Hungar., 123 (2009), 27–39. · Zbl 1199.53166 [3] 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. · Zbl 1084.53039 [4] U. C. De and K. Gazi, On generalized concircularly recurrent manifolds, Studia Math. Hung., 46 (2009), 287–296. · Zbl 1274.53026 [5] U. C. De and N. Guha, On generalized recurrent manifolds, J. Nat. Acad. Math. India, 9 (1991), 85–92. [6] R. S. D. Dubey, Generalized recurrent spaces, Indian J. Pure Appl. Math., 10 (1979), 1508–1513. · Zbl 0422.53007 [7] 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. · Zbl 1181.53031 [8] 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. · Zbl 1085.53036 [9] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. I, Interscience Publishers, a division of John Wiley & Sons (New York–London, 1963). [10] C. A. Mantica and L. G. Molinari, A second-order identity for the Riemann tensor and applications, Colloq. Math., 122 (2011), 69–82. · Zbl 1218.53016 [11] Y. B. Maralabhavi and M. Rathnamma, Generalized recurrent and concircular recurrent manifolds, Indian J. Pure Appl. Math., 30 (1999), 1167–1171. · Zbl 0981.53030 [12] P. J. Ryan, A class of complex hypersurfaces, Colloq. Math., 26 (1972), 175–182. · Zbl 0243.53028 [13] H. S. Ruse, A. G. Walker and T. J. Willmore, Harmonic Spaces, Ed. Cremonese (Roma, 1961). · Zbl 0134.39202 [14] H. Singh and Q. Khan, On generalized recurrent Riemannian manifolds, Publ. Math. Debrecen, 56 (2000), 87–95. · Zbl 0974.53034 [15] L. Vanhecke, Curvature tensors, J. Korean Math. Soc., 14 (1977), 143–151. · Zbl 0367.53006 [16] A. G. Walker, On Ruses’s spaces of recurrent curvature, Proc. London Math. Soc., 52 (1950), 36–64. · Zbl 0039.17702 [17] Y.-C. Wong, Recurrent tensors on a linearly connected differentiable manidfold, Trans. Amer. Math. Soc., 99 (1961), 325–341. · Zbl 0103.38805 [18] Y.-C. Wong, Linear connexions with zero torsion and recurrent curvature, Trans. Amer. Math. Soc., 102 (1962), 471–506. · Zbl 0139.39702 [19] K. Yano, Concircular geometry. I. Concircular transformations, II. Integrability conditions of {$$\mu$$}{$$\nu$$} ={$$\Phi$$}g {$$\mu$$}{$$\nu$$} , 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.
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