Israel Journal of Mathematics

, Volume 178, Issue 1, pp 187–208 | Cite as

Cayley form, comass, and triality isomorphisms

  • Mikhail G. KatzEmail author
  • Steve Shnider


Following an idea of Dadok, Harvey and Morgan, we apply the triality property of Spin(8) to calculate the comass of self-dual 4-forms on ℝ8. In particular, we prove that the Cayley form has comass 1 and that any self-dual 4-form realizing the maximal Wirtinger ratio (equation (1.5)) is SO(8)-conjugate to the Cayley form. We also use triality to prove that the stabilizer in SO(8) of the Cayley form is Spin(7). The results have applications in systolic geometry, calibrated geometry, and Spin(7) manifolds.


Weight Space Maximal Torus Dynkin Diagram Cartan Subalgebra Coadjoint Representation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Ac98]
    B. S. Acharya, On mirror symmetry for manifolds with exceptional holonomy, Nuclear Physics B524 (1998) 269–282, hep-th/9611036.MathSciNetGoogle Scholar
  2. [Ad96]
    J. Adams, Lectures on Exceptional Lie Groups, With a Foreword by J. Peter May, in Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996.Google Scholar
  3. [BCIK07]
    V. Bangert, C. Croke, S. Ivanov and M. Katz, Boundary case of equality in optimal Loewner-type inequalities, Transactions of the American Mathematical Society 359 (2007), 1–17. See arXiv:math.DG/0406008zbMATHCrossRefMathSciNetGoogle Scholar
  4. [BK04]
    V. Bangert and M. Katz, An optimal Loewner-type systolic inequality and harmonic one-forms of constant norm, Communications in Analytical Geometry 12 (2004), 703–732. See arXiv:math.DG/0304494zbMATHMathSciNetGoogle Scholar
  5. [BKSW08]
    V. Bangert, M. Katz, S. Shnider and S. Weinberger, E7, Wirtinger inequalities, Cayley 4-form, and homotopy, Duke Mathematical Journal 146 (2009), 35–70.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [Be96]
    K. Becker, M. Becker, D. R. Morrison, O. Oguri, Y. Oz and Z. Yin, Supersymmetric cycles in exceptional holonomy manifolds and Calabi-Yau 4-folds, Nuclear Physics B480 (1996), 225–238, hep-th/9608116.Google Scholar
  7. [Ber55]
    M. Berger, Sur les groupes d’holonomie homogène des variétés riemanniennes, Bulletin de la Société Mathématique de France 83 (1955), 279–330.zbMATHGoogle Scholar
  8. [Bo66]
    E. Bonan, Sur les variétés riemanniennes `a groupe d’holonomie G 2 ou Spin(7), Comptes Rendus de l’Académie des Sciences, Série A, Sciences Mathématiques 262 (1966), 127–129.zbMATHMathSciNetGoogle Scholar
  9. [BG67]
    R. Brown and A. Gray, Vector cross products, Commentarii Mathematici Helvitici 42 (1967), 222–236.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [Bru08]
    M. Brunnbauer, Homological invariance for asymptotic invariants and systolic inequalities, Geometry and Functional Analysis (GAFA) 18 (2008), 1087–1117.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [Bry87]
    R. Bryant, Metrics with exceptional holonomy, Annales of Mathematics (2) 126 (1987), 525–576.CrossRefGoogle Scholar
  12. [BryH89]
    R. Bryant and F. R. Harvey, Submanifolds in hyperkähler geometry, Journal of the American Mathematical Society 2 (1989), 1–31.zbMATHMathSciNetGoogle Scholar
  13. [BryS89]
    R. Bryant and S. Salamon, On the construction of some complete metrics with exceptional holonomy, Duke Mathematical Journal 58 (1989), 829–850.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [Cu63]
    C. W. Curtis, The four and eight square problem and division algebras, in Studies in Modern Algebra, Vol 2. of MAA Studies in Mathematics, Mathematical Association of America, 1963, pp. 100–125.Google Scholar
  15. [Da85]
    J. Dadok, Polar coordinates induced by the actions of compact Lie groups, Transactions of the American Mathematical Society 288 (1985), 125–137.zbMATHMathSciNetGoogle Scholar
  16. [DH93]
    J. Dadok and F. R. Harvey, Calibrations and Spinors, Acta Mathematica 170 (1993), 83–120.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [DHM88]
    J. Dadok, F. R. Harvey and F. Morgan, Calibrations on8, Transactions of the American Mathematical Society 307 (1988), 1–40.zbMATHMathSciNetGoogle Scholar
  18. [DKR08]
    A. Dranishnikov, M. Katz and Y. Rudyak, Small values of the Lusternik-Schnirelmann category for manifolds, Geometry and Topology 12 (2008), 1711–1727. See arXiv:0805.1527zbMATHCrossRefMathSciNetGoogle Scholar
  19. [Gr83]
    M. Gromov, Filling Riemannian manifolds, Journal of Differetial Geometry 18 (1983), 1–147.zbMATHMathSciNetGoogle Scholar
  20. [Gr96]
    M. Gromov, Systoles and intersystolic inequalities, in Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), Sémin. Congr., 1, Soc. Math. France, Paris, 1996, pp. 291–362. sem-cong 1 Scholar
  21. [Gr99]
    M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics 152, Birkhäuser, Boston, MA, 1999.Google Scholar
  22. [Gr07]
    M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Based on the 1981 French original, with appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates. Reprint of the 2001 English edition. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2007.Google Scholar
  23. [Ha83]
    F. R. Harvey, Calibrated geometries, Proc. Int. Cong. Math, 1983.Google Scholar
  24. [Ha90]
    F. R. Harvey, Spinors and calibrations, Perspectives in Mathematics, 9. Academic Press, Inc., Boston, MA, 1990.Google Scholar
  25. [HL82]
    R. Harvey and H. B. Lawson, Calibrated geometries, Acta Mathematica 148 (1982), 47–157.zbMATHCrossRefMathSciNetGoogle Scholar
  26. [Jo96]
    D. Joyce, Compact 8-manifolds with holonomy Spin(7), Inventiones Mathematicae 123 (1996), 507–522.zbMATHMathSciNetGoogle Scholar
  27. [Jo00]
    D. Joyce, Compact Manifolds with Special Holonomy, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000.Google Scholar
  28. [Jo07]
    D. Joyce, Riemannian Holonomy Groups and Calibrated Geometry, OxfordGraduate Texts in Mathematics, 12, Oxford University Press, Oxford, 2007.zbMATHGoogle Scholar
  29. [Ka95]
    M. Katz, Counterexamples to isosystolic inequalities, Geometriae Dedicata 57 (1995), 195–206.zbMATHCrossRefMathSciNetGoogle Scholar
  30. [Ka07]
    M. Katz, Systolic geometry and topology, With an Appendix by Jake P. Solomon, Mathematical Surveys and Monographs, 137, American Mathematical Society, Providence, RI, 2007.Google Scholar
  31. [KL05]
    M. Katz and C. Lescop, Filling area conjecture, optimal systolic inequalities, and the fiber class in abelian covers, in Geometry, Spectral Theory, Groups, and Dynamics, Contemporary Mathematics 387, Amer. Math. Soc., Providence, RI, 2005, pp. 181–200, See arXiv:math.DG/0412011Google Scholar
  32. [Kl63]
    E. Kleinfeld, A Characterization of the Cayley Numbers, in Studies in Modern Algebra, Vol 2. of MAA Studies in Mathematics, Mathematical Association of America, 1963, pp. 126–143.Google Scholar
  33. [Le02]
    J. -H. Lee and N. C. Leung, Geometric Structures on G 2 and Spin(7)-manifolds, Advances in Theoretical and Mathematical Physics 12 (2009), 1–31.MathSciNetGoogle Scholar
  34. [M88]
    F. Morgan, Area-minimizing surfaces, faces of Grassmannians, and Calibration, American Mathematics Monthly 95 (1988), 813–822.zbMATHCrossRefGoogle Scholar
  35. [Sal89]
    S. Salamon, Riemannian Geometry and Holonomy Groups, Pitman Research Notes in Mathematics, Vol. 201, Longmans Scientific and Technical, Harlow Essex, 1989.zbMATHGoogle Scholar
  36. [Sha95]
    S. L. Shatashvili and C. Vafa, Superstrings and manifolds of exceptional holonomy, Selecta Mathematica 1 (1995), 347–381.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2010

Authors and Affiliations

  1. 1.Department of MathematicsBar Ilan UniversityRamat GanIsrael

Personalised recommendations