Skip to main content
SpringerLink
Log in

Contact 3-Manifolds with Pseudo-parallel Characteristic Jacobi Operator

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this article, we study contact metric 3-manifolds with pseudo-parallel characteristic Jacobi operator. Contact metric 3-manifolds with pseudo-parallel characteristic Jacobi operator are \(M_{\ell }\)-manifolds (contact metric 3-manifolds with vanishing characteristic Jacobi operator) or generalized contact \((\kappa ,\mu ,\nu )\)-spaces. Moreover, we prove that contact metric 3-manifolds with pseudo-parallel characteristic Jacobi operator are classified into four classes. In particular, we give a complete classification of homogeneous contact metric 3-manifolds with proper pseudo-parallel characteristic Jacobi operator.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data availability

Data sharing is not applicable.

References

  1. Blair, D.E.: Two remarks on contact metric structures. Tohoku Math. J. (2) 29(3), 319–324 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  2. Blair, D.E.: On the class of contact metric manifolds with \((3-\tau )\)-structure. Note Mat. 16(1), 99–104 (1996)

    MathSciNet  MATH  Google Scholar 

  3. Blair, D. E.: Riemannian Geometry of Contact and Symplectic Manifolds, Progress in Math. 203, Birkhäuser Boston, Inc., Boston, second edition, (2010)

  4. Blair, D.E., Chen, H.: A classification of 3-dimensional contact metric manifolds with \(Q\phi =\phi Q\) II. Bull. Inst. Math. Acad. Sin. 20(4), 379–383 (1992)

    MathSciNet  MATH  Google Scholar 

  5. Blair, D.E., Koufogiorgos, T., Papantoniou, B.J.: Contact metric manifolds satisfying a nullity condition. Israel J. Math. 91, 189–214 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. Blair, D.E., Koufogiorgos, T., Sharma, R.: A classification of 3-dimensional contact metric manifolds with \(Q\varphi =\varphi Q\). Kodai Math. J. 13(3), 391–401 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  7. Blair, D.E., Ledger, A.J.: Critical associated metrics on contact manifolds II. J. Aust. Math. Soc. Ser. A 41, 404–410 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  8. Calvaruso, G., Perrone, D., Vanhecke, L.: Homogeneity on three-dimensional contact metric manifolds. Israel J. Math. 114, 301–321 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cappelletti Montano, B., Di Terlizzi, L.: Geometric structures associated to a contact metric \((\kappa ,\mu )\)-space. Pac. J. Math. 246(2), 257–292 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chern, S. S., Hamilton, R. S.: On Riemannian metrics adapted to three-dimensional contact manifolds, In: Workshop Bonn 1984 (Bonn, 1984), Lecture Notes in Math. 1111, Springer Verlag, pp. 279–305, (1985)

  11. Cho, J.T., Chun, S.H.: The unit tangent sphere bundle whose characteristic Jacobi operator is pseudo-parallel. Bull. Korean Math. Soc. 53(6), 1715–1723 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cho, J.T., Inoguchi, J.: Pseudo-symmetric contact \(3\)-manifolds. J. Korean Math. Soc. 42(5), 913–932 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cho, J.T., Inoguchi, J.: Pseudo-symmetric contact \(3\)-manifolds II. When is the tangent sphere bundle over a surface pseudo-symmetric? Note Mat. 27(1), 119–129 (2007)

    MathSciNet  MATH  Google Scholar 

  14. Cho, J.T., Inoguchi, J.: Characteristic Jacobi operator on contact Riemannian 3-manifolds. Differ. Geom. Dyn. Syst. 17, 49–71 (2015)

    MathSciNet  MATH  Google Scholar 

  15. Cho, J.T., Inoguchi, J.: Contact 3-manifolds with Reeb flow invariant characteristic Jacobi operator. An. Ştiiņţ. Univ. Al. I. Cuza Mat. (N. S.) 63(3), 665–676 (2017)

    MathSciNet  MATH  Google Scholar 

  16. Cho, J.T., Inoguchi, J., Lee, J.-E.: Pseudo-symmetric contact 3- manifolds III. Colloq. Math. 114(1), 77–98 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gouli-Andreou, F.: On contact metric 3-manifolds with \(R(X, ) =0\). Algebras Groups Geom. 17, 393–400 (2000)

    MathSciNet  MATH  Google Scholar 

  18. Gouli-Andreou, F., Moutafi, E.: Two classes of pseudosymmetric contact metric 3-manifolds. Pac. J. Math. 239(1), 17–37 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gouli-Andreou, F., Moutafi, E.: Three classes of pseudosymmetric contact metric 3-manifolds. Pac. J. Math. 245(1), 57–77 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gouli-Andreou, F., Moutafi, E.: On the concircular curvature of a \((\kappa ,\mu ,\nu )\)-manifold. Pac. J. Math. 269(1), 113–132 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gouli-Andreou, F., Moutafi, E., Xenos, P.J.: On \(3\)-dimensional \((\kappa ,\mu ,\nu )\)-contact metrics. Differ. Geom. Dyn. Syst. 14, 55–68 (2012)

  22. Gouli-Andreou, F., Xenos, P.J.: On \(3\)-dimensional contact metric manifolds with \(\nabla _{\xi }\tau =0\). J. Geom. 62(1–2), 154–165 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  23. Inoguchi, J., Lee, J.-E.: Almost cosymplectic \(3\)-manifolds with pseudo-parallel characteristic Jacobi operator. Int. J. Geom. Methods Mod. Phys. 19(8), Artcle ID 2250119 (2022)

  24. Inoguchi, J., Lee, J.-E.: Almost Kenmotsu \(3\)-manifolds with pseudo-parallel characteristic Jacobi operator. RM 78(2), 1–29 (2023)

    MathSciNet  MATH  Google Scholar 

  25. Inoguchi, J., Lee, J.-E.: On the characteristic Jacobi operator of the unit tangent sphere bundles over surfaces, submitted

  26. Koufogiorgos, T., Markellos, M., Papantoniou, V.: The harmonicity of the Reeb vector field on contact metric 3-manifolds. Pac. J. Math. 234(2), 325–344 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Koufogiorgos, T., Tsichlias, C.: On the existence of a new class of contact metric manifolds. Can. Math. Bull. 43, 440–447 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. Koufogiorgos, T., Tsichlias, C.: Generalized \((\kappa ,\mu )\)-contact metric manifolds with \(\Vert {\rm grad}\,\kappa \Vert =\) constant. J. Geom. 78, 83–91 (2003)

  29. Koufogiorgos, T., Tsichlias, C.: Generalized \((\kappa,\mu )\)-contact metric manifolds with \(\xi \mu =0\). Tokyo J. Math. 31(1), 39–57 (2008)

    Article  MathSciNet  Google Scholar 

  30. Koufogiorgos, T., Tsichlias, C.: Three dimensional contact metric manifolds with vanishing Jacobi operator. Beit. Alg. Geom. 50(2), 563–573 (2009)

    MathSciNet  MATH  Google Scholar 

  31. Loiudice, E., Lotta, A.: Canonical fibrations of contact metric \((\kappa ,\mu )\)-spaces. Pac. J. Math. 300(1), 39–63 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lutz, R.: Sur quelques propriétés des formes différentielles en dimension trois, Thèse, Strasbourg, (1971)

  33. Markellos, M., Papantoniou, V.J.: Biharmonic submanifolds in non-Sasakian contact metric \(3\)-manifolds. Kodai Math. J. 34(1), 144–167 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  34. Martinet, J.: Formes de contact sur les variétés de dimension 3, Proc. Liverpool Singularity Sympos. II. Lect. Notes Math. 209, 142–163 (1971)

    Article  Google Scholar 

  35. Milnor, J.: Curvatures of left invariant metrics on Lie groups. Adv. Math. 21, 293–329 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  36. Perrone, D.: Torsion and critical metrics on contact three-manifolds. Kodai Math. J. 13(1), 88–100 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  37. Perrone, D.: Contact Riemannian manifolds satisfying \(R(X,\xi )\cdot R=0\). Yokohama Math. J. 39, 141–149 (1992)

    MathSciNet  MATH  Google Scholar 

  38. Perrone, D.: Homogeneous contact Riemannian three-manifolds. Illinois J. Math. 13, 243–256 (1997)

    MathSciNet  MATH  Google Scholar 

  39. Perrone, D.: Contact metric manifolds whose characteristic vector field is a harmonic vector field. Differ. Geom. Appl. 20, 367–378 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  40. Smolentsev, N.K.: Space of \(K\)-contact metrics of a three-dimensional manifold. Sib. Math. J. 28, 960–965 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  41. Tanno, S.: Variational problems on contact Riemannian manifolds. Trans. Am. Math. Soc. 314, 349–379 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  42. Thurston, W.M., Winkelnkemper, H.E.: On the existence of contact forms. Proc. Am. Math. Soc. 52(1), 345–347 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  43. Wang, W., Dai, X.: Pseudo-parallel characteristic Jacobi operators on contact metric \(3\) manifolds, J. Math. 2021, Article ID 6148940, 6 pages, (2021)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Hokkaido University, Sapporo, 060-0810, Japan

    Jun-ichi Inoguchi

  2. Department of Mathematics, Chonnam National University, Gwangju, 61186, Korea

    Ji-Eun Lee

Authors
  1. Jun-ichi Inoguchi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ji-Eun Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors wrote the main manuscript text and reviewed the manuscript.

Corresponding author

Correspondence to Ji-Eun Lee.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Publisher's Note

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

The first named author is partially supported by JSPS KAKENHI Grant Number 19K03461, JP23K03081.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Inoguchi, Ji., Lee, JE. Contact 3-Manifolds with Pseudo-parallel Characteristic Jacobi Operator. Mediterr. J. Math. 20, 276 (2023). https://doi.org/10.1007/s00009-023-02474-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-023-02474-3

Keywords

Mathematics Subject Classification

Search

Navigation