Skip to main content
Springer Nature Link
Log in

On Almost Complex Structures on Six-dimensional Products of Spheres

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

In this paper, we discuss almost complex structures on the sphere S6 and on the products of spheres S3 × S3, S1 × S5, and S2 × S4. We prove that all almost complex Cayley structures that naturally appear from their embeddings into the Cayley octave algebra ℂa are nonintegrable. We obtain expressions for the Nijenhuis tensor and the fundamental form 𝜔 for each gauge of the space ℂa and prove the nondegeneracy of the form d𝜔. We show that through each point of a fiber of the twistor bundle over S6, a one-parameter family of Cayley structures passes. We describe the set of U(2) ×U(2)- invariant Hermitian metrics on S3 × S3 and find estimates of the sectional sectional curvature. We consider the space of left-invariant, almost complex structures on S3 × S3 = SU(2) × SU(2) and prove the properties of left-invariant structures that yield the maximal value of the norm of the Nijenhuis tensor on the set of left-invariant, orthogonal, almost complex structures.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. G. Bor and L. Hernandez-Lamoneda, “The canonical bundle of a Hermitian manifold,” Bol. Soc. Mat. Mex., 5, No. 3, 187–198 (1999).

    MathSciNet  MATH  Google Scholar 

  2. A. Borel and J.-P. Serre, “Groupes de Lie et puissance reduites de Steenrod,” Am. J. Math., 75, 409–448 (1953).

    MATH  Google Scholar 

  3. R. Bryant, S.-S. Cherns study of almost-complex structures on the six-sphere, e-print arXiv:1405.3405v1.

  4. R. Bryant, “Submanifolds and special structures on the octonians,” J. Differ. Geom.17, 185–232 (1982).

    MathSciNet  MATH  Google Scholar 

  5. J.-B. Butruille, “Classification des variétés approximativement kähleriennes homogénes,” Ann. Global Anal. Geom., 27, 201–225 (2005); arXiv:math/0612655.

    MathSciNet  MATH  Google Scholar 

  6. E. Calabi, “Construction and properties of some six-dimensional almost-complex manifolds,” Trans. Am. Math. Soc., 87, 407–438 (1958).

    MathSciNet  MATH  Google Scholar 

  7. E. Calabi and B. Eckmann, “A class of compact complex manifolds which are not algebraic,” Ann. Math., 58, 494–500 (1935).

    MathSciNet  MATH  Google Scholar 

  8. E. Calabi and H. Gluck, “What are best almost-complex structures on the 6-sphere?” Proc. Symp. Pure Math., 54, No. 2, 99–108 (1993).

    MathSciNet  MATH  Google Scholar 

  9. L. A. Cordero, M. Fernández, A. Gray, “Lie groups with no left invariant complex structures,” Port. Math., 47, No. 2, 183–190 (1990).

  10. B. Datta and S. Subramanian, “Nonexistence of almost complex structures on products of evendimensional spheres,” Topol. Appl., 36, No. 1, 39–42 (1990).

    MATH  Google Scholar 

  11. N. A. Daurtseva, “On the integrability of almost complex structures on a strictly nearly Kählerian 6-manifold,” Sib. Mat. Zh., 55, No. 1, 61–65 (2014).

    MathSciNet  Google Scholar 

  12. N. A. Daurtseva, “On the variety of almost complex structures,” Mat. Zametki, 78, No. 1, 66–71 (2005).

    MathSciNet  Google Scholar 

  13. N. A. Daurtseva, “On the existence of structures of the class G2 on a strictly nearly Kählerian six-dimensional manifold,” Vestn. Tomsk. Univ. Ser. Mat. Mekh., No. 6 (32), 19–24 (2014).

  14. N. A. Daurtseva, “U(n +1) × U(p + 1)-Hermitian metrics on the manifold S2n+1×S2p+1,” Mat. Zametki, 82, No. 2, 207–223 (2007).

    MathSciNet  Google Scholar 

  15. N. A. Daurtseva, “Invariant complex structures on S3×S3,” Investigated in Russia, 81, 882–887 (2004).

    Google Scholar 

  16. N. A. Daurtseva, “Cayley Structures on S6 as sections of twistor bundles,” Vestn. Novosib. Univ. Ser. Mat. Mekh. Inform., 15, No. 4, 43–49 (2015).

    MATH  Google Scholar 

  17. N. A. Daurtseva, “Norm functional of the Nijenhuis tensor on the set of left-invariant almost complex structures on SU(2) × SU(2) that are orthogonal with respect to the Killing–Cartan metric,” Vestn. Kemerovsk. Univ., 17, No. 1, 156–158 (2004).

    MATH  Google Scholar 

  18. N. A. Daurtseva and N. K. Smolentsev, “On the space of almost complex structures on manifolds,” Vestn. Kemerovsk. Univ., 7, 176–186 (2001).

    MATH  Google Scholar 

  19. G. Etesi, “Complex structure on the six-dimensional sphere from a spontaneous symmetry breaking,” J. Math. Phys., 56, 043508-1–043508-21 (2015).

  20. Y. Euh and K. Sekigawa, Orthogonal almost complex structures on the Riemannian products of even-dimensional round spheres, arXiv:1301.6835v2 [math.DG].

  21. A. Gray, “Vector cross products on manifolds,” Trans. Am. Math. Soc., 141, 465–504 (1969).

    MathSciNet  MATH  Google Scholar 

  22. A. Gray, “Weak holonomy groups,” Math. Z., 123, No. 4, 290–300 (1971).

    MathSciNet  MATH  Google Scholar 

  23. A. Gray and L. M. Harvella, “The sixteen classes of almost Hermitian manifolds and their linear invariants,” Ann. Math. Pura Appl., 123, 35–58 (1980).

    MathSciNet  Google Scholar 

  24. H. Hashimoto, T. Koda, K. Mashimi, K. Sekigawa, “Extrinsic homogeneous almost Hermitian 6-dimensional submanifolds in the octonions,” Kodai Math. J., 30, 297–321 (2007).

    MathSciNet  MATH  Google Scholar 

  25. R. Harvey and H. Lawson, “Calibrated geometries,” Acta Math., 148, 47–157 (1982).

    MathSciNet  MATH  Google Scholar 

  26. N. J. Hitchin, “The geometry of three-forms in six dimensions,” J. Differ. Geom., 55, 547–576 (2000).

    MathSciNet  MATH  Google Scholar 

  27. H. Hopf, “Zur Topologie der komplexen Mannigfaltigkeiten,” in: Studies and Essays Presented to R. Courant, Interscience, New York, (1948), pp. 167–185.

  28. B. F. Kirichenko, “Almost Kählerian structures induced by 3-vector products on six-dimensional submanifolds of the Cayley,” Vestn. Mosk. Univ. Ser. Mat. Mekh., 3, 70–75 (1973).

    Google Scholar 

  29. B. F. Kirichenko, “Classification of Kählerian structures induced by 3-vector products on sixdimensional submanifolds of the Cayley algebra,” Izv. Vyssh. Ucheb. Zaved. Mat., 8, 32–38 (1980).

    MATH  Google Scholar 

  30. B. F. Kirichenko, “Stability of almost Hermitian structures induced by 3-vector products on six-dimensional submanifolds of the Cayley algebra,” Ukr. Geom. Sb., 25, 60–68 (1982).

    MathSciNet  MATH  Google Scholar 

  31. B. F. Kirichenko, “Hermitian geometry of six-dimensional symmetric submanifolds of the Cayley algebra,” Vestn. Mosk. Univ. Ser. Mat. Mekh., 3, 6–13 (1994).

    MathSciNet  MATH  Google Scholar 

  32. Sh. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vols. I, II, Interscience, New York–London (1963, 1969).

  33. C. LeBrun, “Orthogonal complex structures on S6,” Proc. Am. Math. Soc., 101, No. 1, 136–138 (1958).

    Google Scholar 

  34. J.-J. Loeb, M. Manjarin, M. Nicolau, Complex and CR-structures on compact Lie groups associated to Abelian actions, e-print arXiv:math/0610915v1 [math.DG].

  35. A. Morimoto, “Structures complexes sur les Groupes de Lie semi-simples,” C. R. Acad. Sci. Paris, 242, 1101–1103 (1956).

    MathSciNet  MATH  Google Scholar 

  36. P.-A. Nagy, “Nearly K¨ahler geometry and Riemannian foliations,” Asian J. Math., 6, No. 3, 481–504 (2002).

    MathSciNet  MATH  Google Scholar 

  37. N. R. O’Braian and J. H. Rawnsley, “Twistor spaces,” Ann. Global Anal. Geom., 3, No. 1, 29–58 (1985).

    MathSciNet  MATH  Google Scholar 

  38. C. K. Peng and Z. Tang, “Integrability condition on an almost complex structure and its application,” Acta Math. Sinica, Engl. Ser., 21, No. 6, 1459–1464 (2005).

    MathSciNet  MATH  Google Scholar 

  39. R. Reyes Carrión, Some special geometries defined by Lie groups, Ph.D. Thesis, Oxford Univ. (1993).

  40. H. Samelson, “A class of complex analytic manifolds,” Port. Math., 12, 129–132 (1953).

    MathSciNet  MATH  Google Scholar 

  41. T. Sasaki, “Classification of invariant complex structures on SL(2,ℝ) and U(2),” Kumamoto J. Sci. Math., 14, 115–123 (1981).

  42. T. Sasaki, “Classification of invariant complex structures on SL(3,ℝ),” Kumamoto J. Sci. Math., 15, 59–72 (1982).

  43. K. Sekigawa, “Almost complex submanifolds of a six-dimensional sphere,” Kodai Math. J., 6, No. 2, 174–185 (1983).

    MathSciNet  MATH  Google Scholar 

  44. N. K. Smolentsev, “Spaces of Riemannian metrics,” Itogi Nauki Tekh. Ser. Sovr. Mat. Prilozh. Temat. Obzory, 31, pp. 69–146 (2003).

    Google Scholar 

  45. N. K. Smolentsev, “Spaces of Riemannian metrics,” J. Math. Sci., 142, No. 5, 2436–2519 (2007).

    MathSciNet  MATH  Google Scholar 

  46. N. K. Smolentsev, “On almost complex Cayley structures on the sphere S6,” Tr. Rubtsovsk. Indust. Inst., No. 12, 78–85 (2003).

    MATH  Google Scholar 

  47. N. K. Smolentsev, “On almost complex structures on the sphere S6,” Vestn. Kemerovsk. Univ. Ser. Mat., 4, 155–162 (2005).

    Google Scholar 

  48. N. K. Smolentsev, “On almost complex structures on six-dimensional products of spheres,” Uch. Zap. Kazansk. Univ. Ser. Fiz.-Mat. Nauki, 151, No. 4, 116–135 (2004).

    MATH  Google Scholar 

  49. N. K. Smolentsev, “Canonical psuedo-Kählerian metrics on six-dimensional nilpotent Lie groups,” Vestn. Kemerovsk. Univ., 3/1 (47), 155–168 (2011).

  50. D. M. Snow, “Invariant complex structures on reductive Lie groups,” J. Reine Angew. Math., 371, 191–215 (1986).

    MathSciNet  MATH  Google Scholar 

  51. Z. Tang, “Curvature and integrability of an almost Hermitian structure,” Int. J. Math., 17, No. 1, 97 105 (2006).

    MathSciNet  MATH  Google Scholar 

  52. I. Vaisman, “Symplectic twistor spaces,” J. Geom. Phys., 3, No. 4, 507–524 (1986).

    MathSciNet  MATH  Google Scholar 

  53. M. Verbitsky, “An intrinsic volume functional on almost complex 6-manifolds and nearly Kähler geometry,” Pac. J. Math., 236, No. 5, 323–344 (2008).

    MATH  Google Scholar 

  54. H. C. Wang, “Closed manifolds with homogeneous complex structure,” Am. J. Math., 76, 1–37 (1954).

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Kemerovo State University, Kemerovo, Russia

    N. A. Daurtseva & N. K. Smolentsev

Authors
  1. N. A. Daurtseva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. K. Smolentsev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. A. Daurtseva.

Additional information

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 146, Geometry, 2018.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Daurtseva, N.A., Smolentsev, N.K. On Almost Complex Structures on Six-dimensional Products of Spheres. J Math Sci 245, 568–600 (2020). https://doi.org/10.1007/s10958-020-04712-5

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-020-04712-5

Keywords and phrases

AMS Subject Classification

Search

Navigation