Authors:
Carlo Alberto Mantica Physics Department, Università degli Studi di Milano and I.N.F.N., Via Celoria 16, 20133 Milano, Italye-mail: carloalberto.mantica@libero.it

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Luca Guido Molinari Physics Department, Università degli Studi di Milano and I.N.F.N., Via Celoria 16, 20133 Milano, Italye-mail: carloalberto.mantica@libero.it

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Abstract.

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 weaklyZ-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.

  • [1] Adati, T. 1951 On subprojective spaces-III Tohoku Math. J. 3 343358 .

  • [2] Arslan, K. Ezentas, R. Murathan, C. Ozgur, C. 2001 On pseudo Ricci-symmetric manifolds Differ. Geom. Dyn. Syst. 3 15.

  • [3] Besse, A. L. 1987 Einstein Manifolds Springer.

  • [4] Chaki, M. C. 1988 On pseudo Ricci symmetric manifolds Bulg. J. Phys. 15 525531.

  • [5] Chaki, M. C. Kawaguchi, T. 2007 On almost pseudo Ricci symmetric manifolds Tensor (N.S) 68 1014.

  • [6] Chaki, M. C. Koley, S. 1994 On generalized pseudo Ricci symmetric manifolds Period. Math. Hung. 28 123129 .

  • [7] Chaki, M. C. Maity, R. K. 2000 On quasi Einstein manifolds Publ. Math. Debrecen 57 257306.

  • [8] Chaki, M. C. Saha, S. K. 1994 On pseudo-projective Ricci symmetric manifolds Bulg. J. Phys. 21 17.

  • [9] De, U. C. Guha, N. Kamilya, D. 1995 On generalized Ricci recurrent manifolds Tensor (N.S.) 56 312317.

  • [10] De, U. C. De, B. K. 1997 On conformally flat generalized pseudo Ricci symmetric manifolds Soochow J. Math. 23 381389.

  • [11] De, U. C. Gazi, A. K. 2009 On conformally flat almost pseudo Ricci symmetric manifolds Kyungpook Math. J. 49 507520.

  • [12] De, U. C. Ghosh, G. K. 2005 Some global properties of weakly Ricci symmetric manifolds Soochow J. Math. 31 8393.

  • [13] De, U. C. Ghosh, S. K. 2000 On conformally flat pseudo symmetric manifolds Balkan J. Geom. Appl. 5 6164.

  • [14] Felice De, F. Clarke, C. J. S. 1990 Relativity on Curved Manifolds Cambridge University Press.

  • [15] Defever, F. Deszcz, R. 1991 On semi Riemannian manifolds satisfying the condition RR=Q(S,R) Geometry and Topology of Submanifolds, III Leeds May 1990 World Sci. Singapore 108130.

    • Search Google Scholar
    • Export Citation
  • [16] Derdzinski, A. Shen, C. L. 1983 Codazzi tensor fields, curvature and Pontryagin forms Proc. Lond. Math. Soc. 47 1526 .

  • [17] Deszcz, R. 1992 On pseudo symmetric spaces Bull. Soc. Math. Belg., Series A 44 134.

  • [18] Eisenhart, L. P., Non Riemaniann Geometry, reprint Dover Ed. (2005).

  • [19] Gebarowsky, A. 1992 Nearly conformally symmetric warped product manifolds Bull. Inst. Math., Acad. Sin. 20 359371.

  • [20] Jana, S. K. Shaikh, A. A. 2007 On quasi conformally flat weakly Ricci symmetric manifolds Acta Math. Hungar. 115 197214 .

  • [21] Khan, Q. 2004 On recurrent Riemannian manifolds Kyungpook Math. J. 44 269276.

  • [22] Kobayashi, S. Nomizu, K. 1963 Foundations of Differential Geometry 1 Interscience New York.

  • [23] Lovelock, D. and Rund, H., Tensors, differential forms and variational principles, reprint Dover Ed. (1988).

  • [24] Mantica, C. A. Molinari, L. G. 2011 A second order identity for the Riemann tensor and applications Colloq. Math. 122 6982 . arXiv:0802.0650v2 [math.DG], 9 Jul 2009.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [25] Mantica, C. A. and Molinari, L. G., Extended Derdzinski–Shen theorem for the Riemann tensor, arXiv:1101.4157 http://arxiv.org/abs/1101.4157 [math.DG], 21 Jan 2011.

    • Search Google Scholar
    • Export Citation
  • [26] Mishra, R. S. 1984 Structures on a Differentiable Manifold and their Applications Chandrama Prakashan Allahabad.

  • [27] Postnikov, M. M. 2001 Geometry VI, Riemannian Geometry Encyclopaedia of Mathematical Sciences 91 Springer.

  • [28] Shouten, J. A. 1954 Ricci-Calculus 2 Springer Verlag.

  • [29] Roter, W. 1987 On a generalization of conformally symmetric metrics Tensor (N.S.) 46 278286.

  • [30] Singh, H. Khan, Q. 1999 On symmetric manifolds Novi Sad J. Math. 29 301308.

  • [31] Tamássy, L. Binh, T. Q. 1993 On weakly symmetries of Einstein and Sasakian manifolds Tensor (N.S.) 53 140148.

  • [32] Yano, K. 1940 Concircular geometry I, concircular trasformations Proc. Imp. Acad. Tokyo 16 195200 .

  • [33] Yano, K. 1944 On the torseforming direction in Riemannian spaces Proc. Imp. Acad. Tokyo 20 340345 .

  • [34] Yano, K. Sawaki, S. 1968 Riemannian manifolds admitting a conformal transformation group J. Differ. Geom. 2 161184.

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Acta Mathematica Hungarica
Language English
Size B5
Year of
Foundation
1950
Volumes
per Year
3
Issues
per Year
6
Founder Magyar Tudományos Akadémia  
Founder's
Address
H-1051 Budapest, Hungary, Széchenyi István tér 9.
Publisher Akadémiai Kiadó
Springer Nature Switzerland AG
Publisher's
Address
H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible
Publisher
Chief Executive Officer, Akadémiai Kiadó
ISSN 0236-5294 (Print)
ISSN 1588-2632 (Online)

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