Annals of Global Analysis and Geometry

, Volume 42, Issue 2, pp 247–265 | Cite as

Co-calibrated G 2 structure from cuspidal cubics

  • Boris Doubrov
  • Maciej DunajskiEmail author


We establish a twistor correspondence between a cuspidal cubic curve in a complex projective plane, and a co-calibrated homogeneous G 2 structure on the seven-dimensional parameter space of such cubics. Imposing the Riemannian reality conditions leads to an explicit co-calibrated G 2 structure on SU(2, 1)/U(1). This is an example of an SO(3) structure in seven dimensions. Cuspidal cubics and their higher degree analogues with constant projective curvature are characterised as integral curves of certain seventh order ODEs. Projective orbits of such curves are shown to be analytic continuations of Aloff–Wallach manifolds, and it is shown that only cubics lift to a complete family of contact rational curves in a projectivised cotangent bundle to a projective plane.


Co-calibrated G2 Twistor theory Cubic curves 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agricola I., Friedrich T.: On the holonomy of connections with skew-symmetric torsion. Math. Ann. Math. Ann. 328, 711–748 (2004)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Agricola I., Becker-Bender J., Friedrich T.: On the topology and the geometry of SO(3)-manifolds. Ann. Global Anal. Geom. 40, 67–84 (2011)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Aloff S., Wallach N.R.: An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures. Bull. Amer. Math. Soc. 81, 93–97 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Baum, H., Friedrich, T., Grunewald, R., Kath, I.: Twistors and Killing spinors on Riemannian manifolds. Teubner Text. Math. 124 (1991)Google Scholar
  5. 5.
    Bryant R.L.: Metrics with exceptional holonomy. Ann. of Math. (2) 126(3), 525–576 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bryant R.L.: Two exotic holonomies in dimension four, path geometries, and twistor theory. Proc. Sympos. Pure Math. 53, 33–88 (1991)Google Scholar
  7. 7.
    Cabrera F.M., Monar M.D., Swann A.F.: Classification of G 2-structures. J. Lond. Math. Soc. 53, 407–416 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Chalkley, R.: Basic global relative invariants for homogeneous linear differential equations. Memoirs AMS 156(744) (2002)Google Scholar
  9. 9.
    Chiossi S., Salamon S.: The Intrinsic Torsion of SU(3) and G 2 Structures Differential Geometry, pp. 115–133. World Sci. Publ, Valencia (2001)Google Scholar
  10. 10.
    Cvetic M., Gibbons G.W., Lu H, Pope C.N.: Cohomogeneity One Manifolds of Spin(7) and G(2). Holonomy Phys. Rev. D65, 106004 (2002)MathSciNetGoogle Scholar
  11. 11.
    Doubrov B.: Contact trivialization of ordinary differential equations. Differential Geom. Appl. 3, 73–84 (2001)MathSciNetGoogle Scholar
  12. 12.
    Doubrov B.: Generalized Wilczynski invariants for non-linear ordinary differential equations. IMA Vol. Math. Appl. 144, 25–40 (2008)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Doubrov B., Komrakov B.: Classification of homogeneous submanifolds in homogeneous spaces. Lobachevskii J. Math. 3, 19–38 (1999)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Dunajski M.: Solitons, Instantons & Twistors. Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford (2009)Google Scholar
  15. 15.
    Dunajski, M., Godliński, M.: \({GL(2, \mathbb{R})}\) structures, G 2 geometry and twistor theory. Quart. J. Math. (2010). \({{\tt arXiv:1002.3963}}\) Google Scholar
  16. 16.
    Dunajski, M., Sokolov, V.V.: On 7th order ODE with submaximal symmetry. J. Geom. Phys. 61, 1258–1262 (2011). \({{\tt arXiv:1002.1620}}\) Google Scholar
  17. 17.
    Dunajski M., Tod K.P.: Paraconformal geometry of nth order ODEs, and exotic holonomy in dimension four. J. Geom. Phys. 56, 1790–1809 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Godliński M, Nurowski P.: \({GL(2, \mathbb{R})}\) geometry of ODEs. J. Geom. Phys. 60, 991–1027 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Grace J.H., Young A.: The Algebra of Invariants. CUP, Cambridge (1903)zbMATHGoogle Scholar
  20. 20.
    Gran U., Gutowski J., Papadopulos G.: IIB backgrounds with five-form flux. Nucl. Phys. B798, 36–71 (2008)CrossRefGoogle Scholar
  21. 21.
    Gutowski, J., Papadopulos, G.: Heterotic Black Horizons. JHEP (2010)Google Scholar
  22. 22.
    Halphen G.: Sur l’equation différentielle des coniques. Bull. Soc. Math. France 7, 83–85 (1879)MathSciNetGoogle Scholar
  23. 23.
    Harris J.: Algebraic Geometry: A First Course. Springer, Berlin (1995)Google Scholar
  24. 24.
    Hitchin, N.: Complex manifolds and Einstein’s equations, in Twistor Geometry and Non-Linear systems. Springer LNM 970, Doebner, H. & Palev. T (1982)Google Scholar
  25. 25.
    Kodaira K.: On stability of compact submanifolds of complex manifolds. Am. J. Math. 85, 79–94 (1963)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Kryński W.: Paraconformal structures and differential equations. Differential Geom. Appl. 28, 523–531 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Noth G.: Differentialinvarianten und invariante Differentialgleichungen zweier zehngliedriger Gruppen. Leipz. Ber. 56, 19–48 (1904); Diss. LeipzigGoogle Scholar
  28. 28.
    Olver P.J.: Equivalence, Invariants, and Symmetry. CUP, Cambridge (1995)zbMATHGoogle Scholar
  29. 29.
    Olver P.: Classical Invariant Theory. CUP, Cambridge (1999)zbMATHCrossRefGoogle Scholar
  30. 30.
    Ovsienko V., Tabachnikov S.: Projective Differential Geometry Old and New, Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (2005)Google Scholar
  31. 31.
    Penrose R.: Nonlinear gravitons and curved twistor theory. Gen. Relativity Gravitation 7, 31–52 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Reidegeld F.: Spaces admitting homogeneous G 2-structures. Differential Geom. Appl. 28, 301–312 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Salamon S.: Riemannian Geometry and Holonomy Groups. Longman Scientific, Harlow, Essex (1989)zbMATHGoogle Scholar
  34. 34.
    Sylvester, J.J.: Lectures on the Theory of Reciprocants (1888)Google Scholar
  35. 35.
    Vinberg E.B., Gorbatsevich V.V., Onishchik A.L.: Structure of Lie Groups and Lie Algebras. Lie Groups and Lie Algebras, III. Encyclopaedia of Mathematical Sciences. Springer, Berlin (1994)Google Scholar
  36. 36.
    Wilczynski E.J.: Projective Differential Geometry of Curves and Ruled Surfaces. Leipzig, Teubner (1905)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Belarussian State UniversityMinskBelarus
  2. 2.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeUK

Personalised recommendations