General Natural \((\alpha ,\varepsilon )\)-Structures

  • S. L. Druţă-RomaniucEmail author


We study in a unified way the \((\alpha ,\varepsilon )\)-structures of general natural lift type on the tangent bundle of a Riemannian manifold. We characterize the general natural \(\alpha \)-structures on the total space of the tangent bundle of a Riemannian manifold, and provide their integrability conditions (the base manifold is a space form and some involved coefficients are rational functions of the other ones). Then, we characterize the two classes (with respect to the sign of \(\alpha \varepsilon \)) of \((\alpha ,\varepsilon )\)-structures of general natural type on TM. The class \(\alpha \varepsilon =-1\) is characterized by some proportionality relations between the coefficients of the metric and those of the \(\alpha \)-structure, and in this case, the structure is almost Kählerian if and only if the first proportionality factor is the derivative of the second one. Moreover, the total space of the tangent bundle is a Kähler manifold if and only if it depends on three coefficients only (two coefficients of the integrable \(\alpha \)-structure and a proportionality factor).


Natural lift \((\alpha , \varepsilon )\)-Structure Almost Hermitian metric Almost Kähler structure 

Mathematics Subject Classification

Primary 53C15 53B35 53C55 



The author wants to express her gratitude to Professor Fernando Etayo Gordejuela, for carefully reading the paper, and for his valuable suggestions, that led to the improvement of the paper.


  1. 1.
    Abbassi, M.T.K., Sarih, M.: On some hereditary properties of Riemannian \(g\)-natural metrics on tangent bundles of Riemannian manifolds. Diff. Geom. Appl. 22, 19–47 (2005)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alekseevsky, D.V., Medori, C., Tomassini, A.: Homogeneous para-Kahler Einstein manifolds. Russ. Math. Surv. 64(1), 1–43 (2009)CrossRefGoogle Scholar
  3. 3.
    Alegre, P., Carriazo, A.: Slant submanifolds of para-Hermitian manifolds. Mediterr. J. Math. 14(5), 214 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bejan, C.: A classification of the almost parahermitian manifolds. Proc. Conference on Diff. Geom. and Appl., Dubrovnik, 23–27 (1988)Google Scholar
  5. 5.
    Bejan, C.: Almost parahermitian structures on the tangent bundle of an almost paracohermitian manifold. Proc. Fifth Nat. Sem. Finsler and Lagrange spaces, Braşov, 105–109 (1988)Google Scholar
  6. 6.
    Bejan, C.L., Druţă-Romaniuc, S.L.: Harmonic almost complex structures with respect to general natural metrics. Mediterr. J. Math. 11, 123–136 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chursin, M., Schäfer, L., Smoczyk, K.: Mean curvature flow of space-like Lagrangian submanifolds in almost para-Kähler manifolds. Calc. Var. 41(12), 111–125 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Cruceanu, V.: Selected Papers (37. Para-Hermitian and para-Kähler manifolds, pp. 339–387.), Editura PIM, Iaşi (2006)Google Scholar
  9. 9.
    Cruceanu, V., Etayo, F.: On almost para-Hermitian manifolds. Algebras Groups Geom. 16(1), 47–61 (1999)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Druţă-Romaniuc, S.L.: General natural Riemannian almost product and para-Hermitian structures on tangent bundles. Taiwan. J. Math. 16(2), 497–510 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Druţă, S.L., Oproiu, V.: General natural Kähler structures of constant holomorphic sectional curvature on tangent bundles. An. Ştiinţ. Univ. “Al. I. Cuza” Iaşi. Mat. (N.S.) 53(1), 149–166 (2007)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Etayo, F., Santamaria, R.: \((J^2 = \pm 1)-\) metric manifolds. Publ. Math. Debrecen. 57(3–4), 435–444 (2000)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Etayo, F., Santamaria, R.: Distinguished connections on \((J^2 = \pm 1)-\)metric manifolds. Arch. Math. (Brno) 52, 159–203 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Etayo, F., Santamaria, R.: The well adapted connection of a \((J^2 = \pm 1)\)-metric manifold. Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Math. RACSAM 111(2), 355–375 (2017)CrossRefGoogle Scholar
  15. 15.
    Falcitelli, M., Ianus, S., Pastore, A.M.: Linear pseudoconnections on the tangent bundle of a differentiable manifold (Italian). Bull. Math. Soc. Sci. Math. Répub. Soc. Roum., Nouv. Sér. 28(76), 235–249 (1984)zbMATHGoogle Scholar
  16. 16.
    Gadea, P.M., Muñoz Masqué, J.: Classification of almost para-Hermitian manifolds. Rend. Mat. Appl. 7(11), 377–396 (1991)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Ganchev, G., Borisov, A.V.: Note on the almost complex manifolds with a Norden metric. C. R. Acad. Bulgare Sci. 39(5), 31–34 (1986)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Ganchev, G., Mihova, V.: Canonical connection and the canonical conformal group on an almost complex manifold with B-metric. Annuaire Univ. Sofia Fac. Math. Inform. 81(1), 195–206 (1987)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Gray, A., Hervella, L.M.: The sixteen classes of almost Hermitian manifolds and their linear invariants. Ann. Mat. Pura Appl. 123(1), 35–58 (1980)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Ida, C., Manea, A.: On the Integrability of generalized almost para-Norden and para-Hermitian Structures. Mediterr. J. Math. 14(4), 173 (2017)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ivanov, S., Zamkovoy, S.: Parahermitian and paraquaternionic manifolds. Diff. Geom. Appl. 23(2), 205–234 (2005)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Janyška, J.: Natural 2-forms on the tangent bundle of a Riemannian manifold. Rend. Circ. Mat. Palermo, Serie II, Supplemento, The Proceedings of the Winter School Geometry and Topology Srní-January 1992 32, 165–174 (1993)Google Scholar
  23. 23.
    Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. I and II. Interscience, N. York (1963, 1969)Google Scholar
  24. 24.
    Kolář, I., Michor, P., Slovak, J.: Natural Operations in Differential Geometry. Springer-Verlag, Berlin (1993)CrossRefGoogle Scholar
  25. 25.
    Kowalski, O., Sekizawa, M.: Natural transformations of Riemannian metrics on manifolds to metrics on tangent bundles - a classification. Bull. Tokyo Gakugei Univ. 4(40), 1–29 (1988)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Mekerov, D.: On Riemannian almost product manifolds with nonintegrable structure. J. Geom. 89(1–2), 119–129 (2008)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Mekerov, D., Manev, M.: Natural connection with totally skew-symmetric torsion on Riemann almost product manifolds. Int. J. Geom. Methods Modern Physics 9, 1, 14 (2012)zbMATHGoogle Scholar
  28. 28.
    Mihai, I., Nicolau, C.: Almost product structures on the tangent bundle of an almost paracontact manifold. Demonstratio Math. 15, 1045–1058 (1982)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Mok, K.P., Patterson, E.M., Wong, Y.C.: Structure of symmetric tensors of type (0,2) and tensors of type (1,1) on the tangent bundle. Trans. Amer. Math. Soc. 234, 253–278 (1977)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Munteanu, M.I.: Some aspects on the geometry of the tangent bundles and tangent sphere bundles of a Riemannian manifold. Mediterr. J. Math. 5(1), 43–59 (2008)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Naveira, A.M.: A classification of Riemannian almost-product manifolds. Rend. Mat. 3, 577–592 (1983)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Olszak, Z.: Four-dimensional para-Hermitian manifold. Tensor (N. S.) 56, 215–226 (1995)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Oproiu, V.: A generalization of natural almost Hermitian structures on the tangent bundles. Math. J. Toyama Univ. 22, 1–14 (1999)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Oproiu, V.: General natural almost Hermitian and anti-Hermitian structures on the tangent bundles. Bull. Math. Soc. Sc. Math. Roumanie 43(93), 325–340 (2000)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Oproiu, V., Papaghiuc, N.: General natural Einstein Kahler structures on tangent bundles. Diff. Geom. Appl. 27, 384–392 (2009)CrossRefGoogle Scholar
  36. 36.
    Papaghiuc, N.: Some almost complex structures with Norden metric on the tangent bundle of a space form. An Ştiinţ. Univ. “Al. I. Cuza”, Iaşi, Mat. N.S. 46(1), 99–110 (2000)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Peyghan, E., Heydari, A.: A class of locally symmetric para-Kähler Einstein structures on the cotangent bundle. International Mathematical Forum 5(3), 145–153 (2010)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Prvanovic, M.: Holomorphically projective transformations in a locally product space. Math. Balkanika (N.S.) 1, 195–213 (1971)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Rashevsky, P.K.: The scalar field in a stratified space. Trudy Sem. Vektor. Tenzor. Anal. 6, 225–248 (1948)MathSciNetGoogle Scholar
  40. 40.
    Rozenfeld, B.A.: On unitary and stratified spaces. Trudy Sem. Vektor. Tenzor. Anal. 7, 260–275 (1949)MathSciNetGoogle Scholar
  41. 41.
    Staikova, M., Gribachev, K.: Cannonical connections and conformal invariants on Riemannian almost product manifolds. Serdica Math. J. 18, 150–161 (1992)zbMATHGoogle Scholar
  42. 42.
    Teofilova, M.: Almost complex connections on almost complex manifolds with Norden metric. In: Sekigawa, K., Gerdjikov, V.S., Dimiev, S. (eds.) Trends in Differential Geometry, Complex Analysis and Mathematical Physics Proceedings of the 9th International Workshop on Complex Structures, Integrability and Vector Fields, Sofia, Bulgaria, 25–29 August 2008 pp. 231–240. World Scientific, Singapore (2009)Google Scholar
  43. 43.
    Yano, K.: Differential Geometry of Complex and Almost Complex Spaces. Pergamon Press, Oxford (1965)zbMATHGoogle Scholar
  44. 44.
    Yano, K., Ishihara, K.: Tangent and Cotangent Bundles. M. Dekker Inc., New York (1973)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics and InformaticsTechnical University “Gheorghe Asachi” of IaşiIasiRomania

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