Skip to main content
SpringerLink
Log in

Lagrangian submanifolds in the homogeneous nearly Kähler \({\mathbb {S}}^3 \times {\mathbb {S}}^3\)

  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

In this paper, we investigate Lagrangian submanifolds in the homogeneous nearly Kähler \(\mathbb {S}^3 \times \mathbb {S}^3\). We introduce and make use of a triplet of angle functions to describe the geometry of a Lagrangian submanifold in \(\mathbb {S}^3 \times \mathbb {S}^3\). We construct a new example of a flat Lagrangian torus and give a complete classification of all the Lagrangian immersions of spaces of constant sectional curvature. As a corollary of our main result, we obtain that the radius of a round Lagrangian sphere in the homogeneous nearly Kähler \(\mathbb {S}^3 \times \mathbb {S}^3\) can only be \(\frac{2}{\sqrt{3}}\) or \(\frac{4}{\sqrt{3}}\).

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

References

  1. Belgun, F., Moroianu, A.: Nearly Kähler 6-manifolds with reduced holonomy. Ann. Glob. Anal. Geom. 19(4), 307–319 (2001)

    Article  MATH  Google Scholar 

  2. Bolton, J., Dillen, F., Dioos, B., Vrancken, L.: Almost complex surfaces in the nearly Kähler \({\mathbb{S}}^{3}\times {\mathbb{S}}^{3}\). Tôhoku Math. J. 67(1), 1–17 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Butruille, J.-B.: Classification des variétés approximativement kähleriennes homogènes. Ann. Glob. Anal. Geom. 27(3), 201–225 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Butruille, J.-B.: Homogeneous nearly Kähler manifolds. In: Handbook of Pseudo-Riemannian Geometry and Supersymmetry, 399–423, IRMA Lect. Math. Theor. Phys., 16, Eur. Math. Soc., Zürich, (2010)

  5. Castro, I., Torralbo, F., Urbano, F.: On Hamiltonian stationary Lagrangian spheres in non-Einstein Kähler surfaces. Math. Z. 271(1–2), 257–270 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, B.-Y.: Riemannian geometry of Lagrangian submanifolds. Taiwan. J. Math. 5(4), 681–723 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, B.-Y.: Pseudo-Riemannian Submanifolds, \(\delta \)-invariants and Applications. World Scientific Publishing Company, Singapore (2011)

    Book  Google Scholar 

  8. Chen, B.-Y., Ogiue, K.: On totally real submanifolds. Trans. Am. Math. Soc. 193, 257–266 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cleyton, R., Swann, A.: Einstein metrics via intrinsic or parallel torsion. Math. Z. 247(3), 513–528 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dillen, F., Opozda, B., Verstraelen, L., Vrancken, L.: On totally real 3-dimensional submanifolds of the nearly Kaehler 6-sphere. Proc. Am. Math. Soc. 99(4), 741–749 (1987)

    MathSciNet  MATH  Google Scholar 

  11. Dillen, F., Verstraelen, L., Vrancken, L.: Classification of totally real \(3\)-dimensional submanifolds of \({S}^6(1)\) with \({{K}\ge 1/16}\). J. Math. Soc. Jpn. 42(4), 565–584 (1990)

    Article  MATH  Google Scholar 

  12. Dillen, F., Vrancken, L.: Totally real submanifolds in \(S^6(1)\) satisfying Chen’s equality. Trans. Am. Math. Soc. 348(4), 1633–1646 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dioos, B., Li, H.., Ma, H., Vrancken, L.: Flat almost complex surfaces in \({S}^3\times {S}^3\), Preprint, (2015)

  14. Ejiri, N.: Totally real submanifolds in a 6-sphere. Proc. Am. Math. Soc. 83(4), 759–763 (1981)

    MathSciNet  MATH  Google Scholar 

  15. Ejiri, N.: Totally real minimal immersions of \(n\)-dimensional real space forms into \(n\)-dimensional complex space forms. Proc. Am. Math. Soc. 84(2), 243–246 (1982)

    MathSciNet  MATH  Google Scholar 

  16. Foscolo, L., Haskins, M.: New \(G_2\)-holonomy cones and exotic nearly Kähler structures on \({\mathbb{S}}^{6}\) and \({\mathbb{S}}^{3}\times {\mathbb{S}}^{3}\). Ann. of Math. (2) 185(1), 59–130 (2007)

    Article  MATH  Google Scholar 

  17. Gray, A.: Nearly Kähler manifolds. J. Differ. Geom. 4, 283–309 (1970)

    Article  MATH  Google Scholar 

  18. Gray, A.: The structure of nearly Kähler manifolds. Math. Ann. 223(3), 233–248 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gutowski, J., Ivanov, S., Papadopoulos, G.: Deformations of generalized calibrations and compact non-Kähler manifolds with vanishing first Chern class. Asian J. Math. 7, 39–79 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lotay, J.D.: Ruled Lagrangian submanifolds of the 6-sphere. Trans. Am. Math. Soc. 363(5), 2305–2339 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ma, H., Ohnita, Y.: On Lagrangian submanifolds in complex hyperquadrics and isoparametric hypersurfaces in spheres. Math. Z. 261(4), 749–785 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ma, H., Ohnita, Y.: Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces. I. J. Differ. Geom. 97(2), 275–348 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Moroianu, A., Nagy, P.A., Semmelmann, U.: Unit Killing vector fields on nearly Kähler manifolds. Int. J. Math. 16, 281–301 (2005)

    Article  MATH  Google Scholar 

  24. Moroianu, A., Semmelmann, U.: Infinitesimal Einstein deformations of nearly Kähler metrics. Trans. Am. Math. Soc. 363(6), 3057–3069 (2011)

    Article  MATH  Google Scholar 

  25. Moroianu, A., Semmelmann, U.: Generalized Killing spinors and Lagrangian graphs. Differ. Geom. Appl. 37, 141–151 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Moruz, M., Vrancken, L.: Properties of the nearly Kähler \(S^3\times S^3\), Publ. Inst. Math. (Beograd), (to appear)

  27. Nagy, P.-A.: Nearly Kähler geometry and Riemannian foliations. Asian J. Math. 6(3), 481–504 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. Nagy, P.-A.: On nearly-Kähler geometry. Ann. Glob. Anal. Geom. 22(2), 167–178 (2002)

    Article  MATH  Google Scholar 

  29. Oh, Y.-G.: Second variation and stabilities of minimal Lagrangian submanifolds in Kähler manifolds. Invent. Math. 101(2), 501–519 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  30. Oh, Y.-G.: Volume minimization of Lagrangian submanifolds under Hamiltonian deformations. Math. Z. 212(2), 175–192 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  31. Podestà, F., Spiro, A.: 6-dimensional nearly Kähler manifolds of cohomogeneity one. J. Geom. Phys. 60(6), 156–164 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Schäfer, L., Smoczyk, K.: Decomposition and minimality of Lagrangian submanifolds in nearly Kähler manifolds. Ann. Glob. Anal. Geom. 37(3), 221–240 (2009)

    Article  MATH  Google Scholar 

  33. Szabo, Z.I.: Structure theorems on riemannian spaces satisfying \({R}({X},{Y})\cdot {R}=0\). I. the local version. J. Differ. Geom. 17(4), 531–582 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wolf, J.A.: Spaces of Constant Curvature. McGraw-Hill Book Co., New York (1967)

    MATH  Google Scholar 

  35. Zhang, Y., Hu, Z., Dioos, B., Vrancken, L., Wang, X.: Lagrangian submanifolds in the 6-dimensional nearly Kähler manifolds with parallel second fundamental form. J. Geom. Phys. 108, 21–37 (2016)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referee for the very careful review and for providing a number of valuable comments and suggestions.

Author information

Authors and Affiliations

  1. Departement Wiskunde, KU Leuven, Celestijnenlaan 200B, 3001, Leuven, Belgium

    Bart Dioos & Luc Vrancken

  2. LAMAV, Université de Valenciennes, Campus du Mont Houy, 59313, Valenciennes Cedex 9, France

    Luc Vrancken

  3. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, China

    Xianfeng Wang

Authors
  1. Bart Dioos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luc Vrancken
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xianfeng Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xianfeng Wang.

Additional information

X. Wang was supported in part by NSFC Grant No. 11571185.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dioos, B., Vrancken, L. & Wang, X. Lagrangian submanifolds in the homogeneous nearly Kähler \({\mathbb {S}}^3 \times {\mathbb {S}}^3\) . Ann Glob Anal Geom 53, 39–66 (2018). https://doi.org/10.1007/s10455-017-9567-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-017-9567-z

Keywords

Mathematics Subject Classification

Search

Navigation