Advertisement

Journal of Geometry

, 109:6 | Cite as

On metric connections with torsion on the cotangent bundle with modified Riemannian extension

  • Lokman Bilen
  • Aydin Gezer
Article

Abstract

Let M be an n-dimensional differentiable manifold equipped with a torsion-free linear connection \(\nabla \) and \(T^{*}M\) its cotangent bundle. The present paper aims to study a metric connection \(\widetilde{ \nabla }\) with nonvanishing torsion on \(T^{*}M\) with modified Riemannian extension \({}\overline{g}_{\nabla ,c}\). First, we give a characterization of fibre-preserving projective vector fields on \((T^{*}M,{}\overline{g} _{\nabla ,c})\) with respect to the metric connection \(\widetilde{\nabla }\). Secondly, we study conditions for \((T^{*}M,{}\overline{g}_{\nabla ,c})\) to be semi-symmetric, Ricci semi-symmetric, \(\widetilde{Z}\) semi-symmetric or locally conharmonically flat with respect to the metric connection \( \widetilde{\nabla }\). Finally, we present some results concerning the Schouten–Van Kampen connection associated to the Levi-Civita connection \( \overline{\nabla }\) of the modified Riemannian extension \(\overline{g} _{\nabla ,c}\).

Keywords

Cotangent bundle fibre-preserving projective vector field metric connection Riemannian extension semi-symmetry 

Mathematics Subject Classification

53C07 53C35 53A45 

References

  1. 1.
    Aslanci, S., Cakan, R.: On a cotangent bundle with deformed Riemannian extension. Mediterr. J. Math. 11(4), 1251–1260 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Afifi, Z.: Riemann extensions of affine connected spaces. Quart. J. Math. Oxf. Ser. 2(5), 312–320 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Calvino-Louzao, E., García-Río, E., Gilkey, P., Vazquez-Lorenzo, A.: The geometry of modified Riemannian extensions. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465(2107), 2023–2040 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Calviño-Louzao, E., García-Río, E., V ázquez-Lorenzo, R.: Riemann extensions of torsion-free connections with degenerate Ricci tensor. Can. J. Math. 62(5), 1037–1057 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Derdzinski, A.: Connections with skew-symmetric Ricci tensor on surfaces. Results Math. 52(3–4), 223–245 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dryuma, V.: The Riemann extensions in theory of differential equations and their applications. Mat. Fiz. Anal. Geom. 10(3), 307–325 (2003)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Garcia-Rio, E., Kupeli, D.N., Vazquez-Abal, M.E., Vazquez-Lorenzo, R.: Affine Osserman connections and their Riemann extensions. Differ. Geom. Appl. 11(2), 145–153 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gezer, A., Bilen, L., Cakmak, A.: Properties of modified Riemannian extensions. Zh. Mat. Fiz. Anal. Geom. 11(2), 159–173 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hayden, H.A.: Sub-spaces of a space with torsion. Proc. Lond. Math. Soc. S2–34, 27–50 (1932)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ikawa, T., Honda, K.: On Riemann extension. Tensor (N.S.) 60(2), 208–212 (1998)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Ianus, S.: Some almost product structures on manifolds with linear connection. Kodai Math. Sem. Rep. 23, 305–310 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ishii, Y.: On conharmonic transformations. Tensor 7(2), 73–80 (1957)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Kowalski, O., Sekizawa, M.: On natural Riemann extensions. Publ. Math. Debr. 78(3–4), 709–721 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mantica, C.A., Molinari, L.G.: Weakly Z symmetric manifolds. Acta Math. Hung. 135(1–2), 80–96 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mikes, J.: On geodesic mappings of 2-Ricci symmetric Riemannian spaces. Math. Notes 28, 622–624 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Mikes, J.: Geodesic mappings of special Riemannian spaces. Topics in diff. Geometry, Pap. Colloq., Hajduszoboszlo/Hung. 1984, Vol. 2, Colloq. Math. Soc. J. Bolyai 46, North-Holland, Amsterdam, pp. 793–813 (1988).Google Scholar
  17. 17.
    Mikes, J.: Geodesic mappings of affine-connected and Riemannian spaces. J. Math. Sci. (New York) 78(3), 311–333 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mikes, J.: Holomorphically projective mappings and their generalizations. J. Math. Sci. (New York) 89(3), 1334–1353 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mikes, J., Rachunek, L.: T-semisymmetric spaces and concircular vector fields. Suppl. Rend. Circ. Mat. Palermo II. Ser. 69, 187–193 (2002)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Mikes, J., Stepanova, E., Vanzurova, A., et al.: Differential Geometry of Special Mappings. Palacky Univ. Press, Olomouc (2015)zbMATHGoogle Scholar
  21. 21.
    Mok, K.P.: Metrics and connections on the cotangent bundle. Kodai Math. Sem. Rep. 28(2–3), 226–238 (1976/77)Google Scholar
  22. 22.
    Patterson, E.M., Walker, A.G.: Riemann extensions. Quart. J. Math. Oxf. Ser. 2(3), 19–28 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Schouten, J.A., van Kampen, E.R.: Zur Einbettungs- und Krummungstheorie nichtholonomer Gebilde. Math. Ann. 103(1), 752–783 (1930)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Sinjukov, N.S.: Geodesic Mappings of Riemannian Spaces (Russian). Publishing House “Nauka”, Moscow (1979)zbMATHGoogle Scholar
  25. 25.
    Szabo, Z.I.: Structure theorems on Riemannian spaces satisfying \(R(X, Y) \cdot R=0\). I. The local version. J. Differ. Geom. 17, 531–582 (1982)Google Scholar
  26. 26.
    Szabo, Z.I.: Structure theorems on Riemannian spaces satisfying \(R(X, Y)=0\). II. Global version. Geom. Dedic. 19, 65–108 (1985)Google Scholar
  27. 27.
    Toomanian, M.: Riemann extensions and complete lifts of s-spaces. Ph.D. Thesis, The university, Southampton, (1975)Google Scholar
  28. 28.
    Vanhecke, L., Willmore, T.J.: Riemann extensions of D’Atri spaces. Tensor (N.S.) 38, 154–158 (1982)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Willmore, T.J.: Riemann extensions and affine differential geometry. Results Math. 13(3–4), 403–408 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Yano, K., Ishihara, S.: Tangent and Cotangent Bundles. Marcel Dekker Inc., New York (1973)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Computer, Faculty of Science and LettersIgdir UniversityIgdirTurkey
  2. 2.Department of Mathematics, Faculty of ScienceAtaturk UniversityErzurumTurkey

Personalised recommendations