Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-20T06:25:47.488Z Has data issue: false hasContentIssue false
  • English
  • Français

On the twistor space of the six-sphere

Published online by Cambridge University Press:  17 April 2009

Emilio Musso
Emilio Musso
Affiliation:
Dipartimento di Matematica Pure ed Applicata, Universita Dell-Aquila, via Roma 33, 67–100 L'aquila, Italy
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The set of all complex lines of the right-handed Dirac spinor bundle of a standard six-sphere is the total space of the twistor fibration. The twistor space, endowed with its natural Kähler structure, is recognised to be a six-dimensional complex quadric. The relevant group is Spin(7), which acts transitively on the six-quadric, as a group of fiber-preserving isometries. We use a result due to Berard-Bérgery and Matsuzawa to show the existence of a non-Kähler, non symmetric, Hermitian-Einstein metric on the six-quadric, which is Spin(7)-invariant.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 39 , Issue 1 , February 1989 , pp. 119 - 127
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Berard-Bérgery, L.. (unpublished).Google Scholar
[2]Besse, A.L., Einstein manifolds: Ergeb. Math. Grenzgeb. 3. Folge, Bd. 10 (Springer-Verlag, Berlin, Heidelberg, New York, 1987).CrossRefGoogle Scholar
[3]Bourguignon, J.P. and Karcher, H., ‘Curvature operators: pinching estimates and geometric examples’, Ann. Sc. Ecde. Norm. Sup. 11 (1978), 71–92.Google Scholar
[4]Bryant, R., ‘Submanifolds and special structures on the octonians’, J. Differential Geometry 17 (1982), 185–232.CrossRefGoogle Scholar
[5]Bryant, R., ‘Explicit metrics with holonomy G 2 and Spin (7)’. (I.H.E.S., Bures-sur-Yvette, 1985) (preprint).Google Scholar
[6]Eells, and Salamōn, , ‘Twistorial construction of harmonic maps of surfaces into four-manifolds’, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), 589–640.Google Scholar
[7]Gray, A., ‘Vector cross products on manifolds’, Trans. Amer. Math. Soc. 141 (1969), 465–504.CrossRefGoogle Scholar
[8]Jensen, G.R., ‘Einstein metrics on principal fibre bundles’, J. Differential Geom. 8 (1973), 599–614.CrossRefGoogle Scholar
[9]Matsuzawa, T., ‘Einstein metrics on Fibered Riemannian structures’, Kodai Math. J. 6 (1983), 340–345.CrossRefGoogle Scholar
[10]Matsushima, Y., ‘Remarks on Kähler-Einstein manifolds’, Nagoya Math. J. 46 (1972), 161–173.CrossRefGoogle Scholar
[11]Musso, E., Pseudo-holomorphic curves in the six-sphere (Ph.D Thesis, Washington University, 1987).Google Scholar
[12]Wang, M. and Ziller, W., ‘On normal homogeneous Einstein manifolds’, Ann. Sci. Ecole. Norm. Sup. 18 (1985), 563–633.CrossRefGoogle Scholar
[13]Wong, P.M., ‘Twistor spaces over 6-dimensional Riemannian manifolds’, Illinois J. Math. 31 (1987), 274–311.CrossRefGoogle Scholar
[14]Ziller, W., ‘Homogeneous Einstein metrics on spheres and projective spaces’, Math. Ann. 259 (1982), 351–368.CrossRefGoogle Scholar