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On operators associated with tensor fields

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An Erratum to this article was published on 01 December 2010

Abstract

The aim of this paper is to introduce some operators which are applied to pure tensor fields. In this context Tachibana, Vishnevskii, Yano–Ako operators and their generalizations can be found.

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References

  1. Gadea P.M., Grifone J., Munoz Masque J.: Stein embedding theorem for B-manifolds. Proc. Edinb. Math. Soc. 46, 489–499 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ganchev G.T., Borisov A.V.: Note on the almost complex manifolds with Norden metric. C. R. Acad. Bulg. Sci. 39, 31–34 (1986)

    MathSciNet  MATH  Google Scholar 

  3. Gezer A., Salimov A.A.: Diagonal lifts of tensor fields of type (1,1) on cross-sections in tensor bundles and its applications. J. Korean Math. Soc. 45, 367–376 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Kobayashi S., Nomizu K.: Foundations of Differential Geometry, vol. I. Interscience Publishers, New York (1963)

    Google Scholar 

  5. Kobayashi S., Nomizu K.: Foundations of Differential Geometry, vol. II. Interscience Publishers, New York (1969)

    MATH  Google Scholar 

  6. Kruchkovich G.I.: Conditions for the integrability of a regular hypercomplex structure on a manifold (Russian). Ukrain. Geometr. Sb. Vyp. 9, 67–75 (1970)

    MathSciNet  MATH  Google Scholar 

  7. Kruchkovich G.I.: Hypercomplex structures on manifolds I (Russian). Trudy Sem. Vektor. Tenzor. Anal. 16, 174–201 (1972)

    MathSciNet  Google Scholar 

  8. Kruchkovich G.I.: Hypercomplex structures on manifolds II (Russian). Trudy Sem. Vektor. Tenzor. Anal. 17, 218–227 (1974)

    MathSciNet  Google Scholar 

  9. Frolicher, A., Nijenhuis, A.: Some new cohomology invariants for complex manifolds I, II. Nederl. Akad. Wetensch. Proc. Ser. A. 59, 540–552 (1956) (Indag. Math. 18, 553–564)

  10. Frolicher, A., Nijenhuis, A.: Theory of vector-valued differential forms. I. Derivations of the graded ring of differential forms. Nederl. Akad. Wetensch. Proc. Ser. A. 59 (1956) (Indag. Math. 18, 338–359)

  11. Iscan M., Salimov A.A.: On Kähler–Norden manifolds. Proc. Indian Acad. Sci. (Math. Sci.) 119, 71–80 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Magden A., Salimov A.A.: Horizontal lifts of tensor fields to sections of the tangent bundle. Transl. Russ. Math. (Iz. VUZ) 45, 73–76 (2001)

    MathSciNet  Google Scholar 

  13. Magden A., Salimov A.A.: On applications of the Yano–Ako operator. Palack. Olomuc. Fac. Rerum Natur. Math. 45, 135–141 (2006)

    MathSciNet  MATH  Google Scholar 

  14. Magden A., Salimov A.A.: Complete lifts of tensor fields on a pure cross-section in the tensor bundle. J. Geom. 93, 128–138 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Nijenhuis, A.: X n-1-forming sets of eigenvectors. Nederl. Akad. Wetensch. Proc. Ser. A. 54, Indag. Math. 13, 200–212 (1951)

    Google Scholar 

  16. Norden A.P.: On a class of four-dimensional A-spaces (Russian). Izv. Vyssh. Uchebn. Zaved. Matematika. 17, 145–157 (1960)

    MathSciNet  Google Scholar 

  17. Salimov A.A.: Quasiholomorphic mapping and a tensor bundle. Transl. Soviet Math. (Iz. VUZ) 33, 89–92 (1989)

    MathSciNet  MATH  Google Scholar 

  18. Salimov A.A.: Almost ψ-holomorphic tensors and their properties. Transl. Russ. Acad. Sci. Dokl. Math. 45, 602–605 (1992)

    MathSciNet  Google Scholar 

  19. Salimov A.A.: A new method in the theory of liftings of tensor fields in a tensor bundle. Transl. Russ. Math. (Iz. VUZ) 38, 67–73 (1994)

    MathSciNet  Google Scholar 

  20. Salimov A.A.: Generalized Yano–Ako operator and the complete lift of tensor fields. Tensor (N.S.) 55, 142–146 (1994)

    MathSciNet  MATH  Google Scholar 

  21. Salimov A.A.: Lifts of poly-affinor structures on pure sections of a tensor bundle. Transl. Russ. Math. (Iz. VUZ) 40, 52–59 (1996)

    MathSciNet  MATH  Google Scholar 

  22. Salimov A.A.: Nonexistence of Para-Kahler–Norden warped metrics. Int. J. Geom. Methods Mod. Phys. 6, 1097–1102 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Salimov A.A., Magden A.: Complete lifts of tensor fields on a pure cross-section in the tensor bundle \({T_{q}^{1} (M_{n} )}\) . Note Mat. 18, 27–37 (1998)

    MathSciNet  MATH  Google Scholar 

  24. Salimov A.A., Iscan M., Etayo F.: Paraholomorphic B-manifold and its properties. Topol. Appl. 154, 925–933 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Salimov A.A., Iscan M., Akbulut K.: Some remarks concerning hyperholomorphic B-manifolds. Chin. Ann. Math. Ser. B 29, 631–640 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Salimov A.A., Akbulut K.: A note on a paraholomorphic Cheeger–Gromoll metric. Proc. Indian Acad. Sci. (Math. Sci.) 119, 187–195 (2009)

    MathSciNet  MATH  Google Scholar 

  27. Salimov A.A., Iscan M.: Some properties of Norden–Walker metrics. Kodai Math. J. 33, 283–293 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Salimov, A.A., Iscan, M.: On Norden–Walker 4-manifolds. Note Mat. 30(1) (2010, to appear)

  29. Salimov A.A., Agca F.: On para-Nordenian structures. Ann. Polon. Math. 99, 193–200 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Sato I.: Almost analytic tensor fields in almost complex manifolds. Tensor (N.S.) 17, 105–119 (1966)

    MathSciNet  MATH  Google Scholar 

  31. Sekizawa M.: A note on the complete lift of \({\Phi}\) -tensors to a tangent bundle. TRU Math. 5, 43–45 (1969)

    MathSciNet  MATH  Google Scholar 

  32. Shirokov A.P.: On a property of covariantly constant affinors (Russian). Dokl. Akad. Nauk SSSR (N.S.) 102, 461–464 (1955)

    MathSciNet  MATH  Google Scholar 

  33. Shirokov A.P.: On the question of pure tensors and invariant subspaces in manifolds with almost algebraic structure (Russian). Kazan. Gos. Univ. Ueen. Zap. 126, 81–89 (1966)

    MATH  Google Scholar 

  34. Slebodzinski W.: Contribution la gomtrie diffrentielle d’un tenseur mixte de valence deux. Colloq. Math. 13, 49–54 (1964)

    MathSciNet  MATH  Google Scholar 

  35. Tachibana S.: Analytic tensor and its generalization. Tohoku Math. J. 12, 208–221 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  36. Tachibana S., Koto S.: On almost-analytic functions, tensors and invariant subspaces. Tohoku Math. J. 14, 177–186 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  37. Vishnevskii V.V.: On the complex structure of B-spaces (Russian). Kazan. Gos. Univ. Uchen. Zap. 123, 24–48 (1963)

    Google Scholar 

  38. Vishnevskii V.V.: Certain properties of differential geometric structures that are determined by algebras (Russian). Vis Ped. Inst. Plovdiv Naun. Trud. 10(1), 23–30 (1972)

    Google Scholar 

  39. Vishnevskii, V.V.: Affinor structures of manifolds as structures definable by algebras (Survey article, Russian). Commemoration volumes for Prof. Dr. Akitsugu Kawaguchi’s seventieth birthday, vol. III. Tensor (N.S.) 26, 363–372 (1972)

  40. Vishnevskii, V.V., Shirokov, A.P., Shurygin, V.V.: Spaces over algebras (Russian). Kazanskii Gosudarstvennii Universitet, Kazan (1985)

  41. Vishnevskii V.V.: Integrable affinor structures and their plural interpretations. Geometry, 7. J. Math. Sci. (New York) 108, 151–187 (2002)

    Article  MathSciNet  Google Scholar 

  42. Willmore T.J.: Note on the Slebodzinski tensor of an almost-complex structure. J. Lond. Math. Soc. 43, 321–322 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  43. Yano, K.: Differential geometry on complex and almost complex spaces. In: International series of monographs in pure and applied mathematics, vol. 49. Pergamon Press, New York (1965)

  44. Yano K., Ako M.: On certain operators associated with tensor fields. Kodai Math. Sem. Rep. 20, 414–436 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  45. Yano, K., Kon, M.: Structures on manifolds. In: Series in pure mathematics, vol. 3. World Scientific Publishing Co., Singapore (1984)

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Ataturk University, 25240, Erzurum, Turkey

    Arif Salimov

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  1. Arif Salimov
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Correspondence to Arif Salimov.

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This paper is supported by The Scientific and Technological Research Council of Turkey (TBAG-108T590).

An erratum to this article can be found at http://dx.doi.org/10.1007/s00022-011-0070-6

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Salimov, A. On operators associated with tensor fields. J. Geom. 99, 107–145 (2010). https://doi.org/10.1007/s00022-010-0059-6

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  • DOI: https://doi.org/10.1007/s00022-010-0059-6

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