Skip to main content

Generalized Almost Hermitian Spaces and Holomorphically Projective Mappings


We derive some interesting properties of generalized almost Hermitian spaces. Additionally, we study basic equations of holomorphically projective mappings between generalized Kähler spaces and examine two non-linear PDE systems for the existence of these mappings. In the case of an equitorsion holomorphically projective mapping, we transform those non-linear PDE-systems into linear PDE systems. Some new types of generalized Kähler spaces that admit equitorsion holomorphically projective mappings and transformations are defined as a natural consequence of the linear PDE systems for the existence of these diffeomorphisms. Our consideration provides new results as well as extensions and improvements of some results previously published in several papers.

This is a preview of subscription content, access via your institution.


  1. Belova, O., Mikeš, J., Strambach, K.: Complex curves as lines of geometries. Result Math. 71(1–2), 145–165 (2017)

    Article  MathSciNet  Google Scholar 

  2. Domashev, V.V., Mikeš, J.: Theory of holomorphically projective mappings of Kählerian spaces. Mat. Zametki 23(2), 145–165 (1978)

    MATH  Google Scholar 

  3. Eisenhart, L.P.: Generalized Riemannian spaces I. Proc. Natl. Acad. Sci. USA 37, 311–315 (1951)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Hall, G.S., Lonie, D.P.: The principle of equivalence and cosmological metrics. J. Math. Phys. 49, 022502 (2008)

    Article  MathSciNet  Google Scholar 

  6. Hall, G.S., Lonie, D.P.: Projective equivalence of Einstein spaces in general relativity. Class. Quantum Gravity 26, 125009 (2009)

    Article  MathSciNet  Google Scholar 

  7. Hinterleitner, I., Mikeš, J.: On \(F\)-planar mappings of spaces with affine connections. Note Mat. 27, 111–118 (2007)

    MathSciNet  MATH  Google Scholar 

  8. Hinterleitner, I., Mikeš, J., Stránská, J.: Infinitesimal \(F\)-planar transformations, Russ. Math., vol. 52, No. 4, (2008), 13–18. Transl. from Iz. VUZ Matematika, no. 4, 16–21 (2008)

  9. Hinterleitner, I., Mikeš, J.: On holomorphically projective mappings from manifolds with equiaffine connection onto Kähler manifolds. Arch. Math. (Brno) 49, 255–264 (2013)

    MATH  Google Scholar 

  10. Ishihara, S.: Holomorphically projective changes and their groups in an almost complex manifold. Tohoku Math. J. II 9(3), 273–297 (1957)

    Article  MathSciNet  Google Scholar 

  11. Ishihara, S., Tachibana, S.-I.: A note on holomorphically projective transformations of a Kählerian space with parallel Ricci tensor. Tohoku Math. J. II 13(2), 193–200 (1961)

    Article  Google Scholar 

  12. Ivanov, S.: Connections with torsion, parallel spinors and geometry of Spin (7)-manifolds. Math. Res. Lett. 11, 171–186 (2004)

    Article  MathSciNet  Google Scholar 

  13. Kamber, F.W., Tondeur, Ph: Flat manifolds with parallel torsion. J. Differ. Geom. 2, 385–389 (1968)

    Article  MathSciNet  Google Scholar 

  14. Manev, M.: A connection with parallel torsion on almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics. J. Geom. Phys 61(1), 248–259 (2011)

    Article  MathSciNet  Google Scholar 

  15. Mikeš, J.: On holomorphically projective mappings of Kählerian spaces. Ukr. Geom. Sb. 23, 90–98 (1980)

    MATH  Google Scholar 

  16. Mikeš, J.: Holomorphically projective mappings and their generalizations. J. Math. Sci. 89(3), 1334–1353 (1998)

    Article  MathSciNet  Google Scholar 

  17. Mikeš, J., Chudá, H., Hinterleitner, I.: Conformal holomorphically projective mappings of almost Hermitian manifolds with a certain initial condition. Int. J. Geom. Methods Mod. Phys. 11(5), 1450044 (2014)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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

  20. Minčić, S.M.: New commutation formulas in the non-symmetric affine connexion space. Publ. Inst. Math. New Ser. 22(36), 189–199 (1977)

    MathSciNet  MATH  Google Scholar 

  21. Minčić, S.M., Stanković, M.S., Velimirović, LjS: Generalized Kählerian spaces. Filomat 15, 167–174 (2001)

    MATH  Google Scholar 

  22. Minčić, S.M., Velimirović, LjS, Stanković, M.S.: Infinitesimal deformations of a non-symmetric affine connection space. Filomat 15, 175–182 (2001)

    MathSciNet  MATH  Google Scholar 

  23. Petrović, M.Z., Stanković, M.S.: A note on F-planar mappings of manifolds with non-symmetric linear connection. Int. J. Geom. Methods Mod. Phys. 16(5), 1950078 (2019)

    Article  MathSciNet  Google Scholar 

  24. Petrović, M.Z., Velimirović, L.S.: Generalized Kähler spaces in Eisenhart’s sense admitting a holomorphically projective mapping. Mediterr. J. Math. (2018).

    Article  MATH  Google Scholar 

  25. Prvanović, M.: Four curvature tensors of non-symmetric affine connexion (in Russian). In: Proceedings of the All-Union Scientific Conference on Non-Euclidean Geometry “150 years of Lobachevsky geometry”, Kazan, 30th June–2nd July 1976, pp 199–205. VINITI, Moscow (1977)

  26. Prvanović, M.: A note on holomorphically projective transformations of the Kähler spaces. Tensor New Ser. 35, 99–104 (1981)

    MATH  Google Scholar 

  27. Prvanović, M.: Einstein connection of almost Hermitian manifold. Bull. Cl. Sci. Math. Nat. Sci. Math. 20, 51–59 (1995)

    MathSciNet  MATH  Google Scholar 

  28. Velimirović, LjS, Minčić, S.M., Stanković, M.S.: Infinitesimal rigidity and flexibility of a non-symmetric affine connection space. Eur. J. Comb. 31(4), 1148–1159 (2010)

    Article  MathSciNet  Google Scholar 

  29. Yano, K.: Differential Geometry of Complex and Almost Complex Spaces. Pergamon Press, New York (1965)

    MATH  Google Scholar 

Download references


This work was supported by Grant no. 174012 of the Ministry of Education, Science and Technological Development, Republic of Serbia.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Miloš Z. Petrović.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Petrović, M.Z., Velimirović, L.S. Generalized Almost Hermitian Spaces and Holomorphically Projective Mappings. Mediterr. J. Math. 17, 74 (2020).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI:


  • Generalized Kähler space
  • PDE system
  • holomorphically projective mapping
  • torsion tensor
  • infinitesimal transformation

Mathematics Subject Classification

  • Primary 53B05
  • Secondary 53B20
  • 53B35