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Projective Curvature Tensors of Some Special Manifolds with Non-symmetric Linear Connection

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Abstract

We consider geodesic mappings between some special manifolds with the non-symmetric linear connection. We obtain some tensors that are invariant with respect to geodesic mappings by relaxing the so-called “equitorsion condition”. Derived tensors have algebraic expressions analogous to the Weyl tensor of projective curvature, apart from that the generalized Ricci and curvature tensors appeared instead of the usual Ricci and curvature tensors. Further, we study some properties of mentioned invariant tensors as well as their relations with the Weyl tensor of projective curvature. Finally, we introduce some kinds of recurrent manifolds with the non-symmetric linear connection as well as their generalized Ricci and projective counterparts.

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References

  1. Bejan, C.L., Druta-Romaniuc, S.L.: The projective curvature of the tangent bundle with natural diagonal metric. Filomat 29(3), 401–410 (2015)

    Article  MathSciNet  Google Scholar 

  2. Đorić, M., Đorić, M., Moruz, M.: Geodesic lines on nearly Kähler \({\mathbb{S}}^3\times {\mathbb{S}}^3\). J. Math. Anal. Appl. 466(1), 1099–1108 (2018)

    Article  MathSciNet  Google Scholar 

  3. Hall, G.S.: Einstein’s geodesic postulate, projective relatedness and Weyl’s projective tensor, chapter. In: Proceedings of the 14th Regional Conference, Mathematical Physics, pp. 27–35 (2018)

  4. Hall, G.S.: Projective relatedness and conformal flatness. Open Math. 10(5), 1763–1770 (2012)

    MathSciNet  MATH  Google Scholar 

  5. Hall, G.S., Lonie, D.P.: Holonomy and projective equivalence in 4-dimensional Lorentz manifolds. SIGMA 5, 066 (2009)

    MathSciNet  MATH  Google Scholar 

  6. Hall, G.S., Lonie, D.P.: The principle of equivalence and projective structure in spacetimes. Class. Quantum Gravity 24(14), 3617–3636 (2007)

    Article  MathSciNet  Google Scholar 

  7. Hall, G., Wang, Z.: Projective structure in 4-dimensional manifolds with positive definite metrics. J. Geom. Phys. 62, 449–463 (2012)

    Article  MathSciNet  Google Scholar 

  8. Haesen, S., Verstraelen, L.: Natural intrinsic geometrical symmetries. SIGMA 5, 086 (2009)

    MathSciNet  MATH  Google Scholar 

  9. Ivanov, S.: On dual-projectively flat affine connections. J. Geom. 53, 89–99 (1995)

    Article  MathSciNet  Google Scholar 

  10. Mikeš, J.: Geodesic mappings of affine-connected and Riemannian spaces. J. Math. Sci. N. Y. 78(3), 311–333 (1996)

    Article  MathSciNet  Google Scholar 

  11. Mikeš, J., Berezovski, V.E., Stepanova, E., Chudá, H.: Geodesic mappings and their generalizations. J. Math. Sci. 217(5), 607–623 (2016)

    Article  MathSciNet  Google Scholar 

  12. Mikeš, J., et al.: Differential Geometry of Special Mappings. Palacký University Press, Olomouc (2015)

    MATH  Google Scholar 

  13. Minčić, S.M.: Independent curvature tensors and pseudotensors of spaces with non-symmetric affine connexion. In: Colloquia Mathematica Societatis János Bolayai, 31, Differential Geometry, Budapest (Hungary), pp. 445–460 (1979)

  14. Minčić, S.M.: Some characteristics of curvature tensors of non-symmetric affine connexion, 12th Yugoslav geometric seminar (Novi Sad, 1998). Novi Sad J. Math. 6(3), 169–183 (1999)

    Google Scholar 

  15. Minčić, S.M.: On Ricci type identities in manifolds with non-symmetric affine connection. Publ. Inst. Math. (Beograd) (N.S.) 94(108), 205–217 (2013)

    Article  MathSciNet  Google Scholar 

  16. Minčić, S.M., Stanković, M.S.: Equitorsion geodesic mappings of generalized Riemannian spaces. Publ. Inst. Math. (Beograd) (N.S.) 61(75), 97–104 (1997)

    MathSciNet  MATH  Google Scholar 

  17. Minčić, S.M., Stanković, M.S.: On geodesic mappings of general affine connection spaces and of generalized Riemannian spaces. Mat. Vesnik 49(1), 27–33 (1997)

    MathSciNet  MATH  Google Scholar 

  18. Minčić, S.M., Stanković, M.S., Velimirović, L.S.: Generalized Riemannian Spaces and Spaces of Non-symmetric Affine Connection. University of Niš, Faculty of Sciences and Mathematics, Niš (2013)

    MATH  Google Scholar 

  19. Petrović, M.Z., Stanković, M.S.: Special almost geodesic mappings of the first type of non-symmetric affine connection spaces. Bull. Malays. Math. Sci. Soc. (2) 40(3), 1353–1362 (2017)

    Article  MathSciNet  Google Scholar 

  20. Petrović, M.Z., Stanković, M.S.: On almost geodesic mappings of the second type between manifolds with non-symmetric linear connection. Filomat 32(11), 3831–3841 (2018)

    Article  MathSciNet  Google Scholar 

  21. Petrović, M.Z., Stanković, M.S., Peška, P.: On conformal and concircular diffeomorphisms of Eisenhart’s generalized Riemannian spaces. Mathematics 7, 626 (2019). https://doi.org/10.3390/math7070626

    Article  Google Scholar 

  22. Prvanović, M.: Four curvature tensors of non-symmetric affine connexion. In: Proceedings of the Conference on “150 years of Lobachevski geometry”, Kazan 1976, Moscow 1977, pp. 199–205 (in Russian)

  23. Rashevsky, P.K.: On geometry of homogeneous spaces. Dokl. AN SSSR 80, 169–171 (1951). ((in Russian))

    MathSciNet  Google Scholar 

  24. Stanković, M.S., Zlatanović, M.. Lj.., Vesić, N.O.: Some properties of ET-projective tensors obtained from Weyl projective tensor. Filomat 29(3), 573–584 (2015)

    Article  MathSciNet  Google Scholar 

  25. Stanković, M.S., Minčić, S.M., Velimirović, L.S., Zlatanović, M.L.: On equitorsion geodesic mappings of general affine connection spaces. Rend. Semin. Mat. Univ. Padova 124, 77–90 (2010)

    Article  MathSciNet  Google Scholar 

  26. Steglich, Ch.: Invariants of conformal and projective structures. Result Math. 27, 188–190 (1995)

    Article  MathSciNet  Google Scholar 

  27. Stepanov, S.E.: Some conformal and projective scalar invariants of Riemannian manifolds. Math. Notes 80(6), 848–852 (2006)

    Article  MathSciNet  Google Scholar 

  28. Vesić, N.O.: Weyl projective objects \(\underset{1}{\cal{W}}\), \(\underset{2}{{\cal{W}}}\), \(\underset{3}{{\cal{W}}}\) for equitorsion geodesic mappings. Miskolc Math. Notes 19(1), 665–675 (2018)

    Article  MathSciNet  Google Scholar 

  29. Vesić, N.O.: Zlatanović, M. Lj., Velimirović, A. M. : Projective invariants for equitorsion geodesic mappings of semi-symmetric affine connection spaces. J. Math. Anal. Appl. 472(2), 1571–1580 (2019)

  30. Zlatanović, M.L.: New projective tensors for equitorsion geodesic mappings. Appl. Math. Lett. 25, 890–897 (2012)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors wish to express their gratitude to referees for several useful comments that improved the readability of the paper. The research by Miloš Z. Petrović leading to these results was partially supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia, project no. 174012.

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  1. University of Niš, Niš, Serbia

    Miloš Z. Petrović & Ana M. Velimirović

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  1. Miloš Z. Petrović
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  2. Ana M. Velimirović
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Correspondence to Miloš Z. Petrović.

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Dedicated to Professor Svetislav Minčić on the occasion of his 90th birthday.

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Petrović, M.Z., Velimirović, A.M. Projective Curvature Tensors of Some Special Manifolds with Non-symmetric Linear Connection. Mediterr. J. Math. 18, 124 (2021). https://doi.org/10.1007/s00009-021-01768-8

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  • DOI: https://doi.org/10.1007/s00009-021-01768-8

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