Cohomogeneity-one \(G_{2}\)-structures. (English) Zbl 1025.53024

Let \(M\) be a \(7\)-dimensional manifold equipped with a \(G_2\)-structure \(\phi\), which is a certain \(3\)-form on \(M\) that induces an identification of each tangent space of \(M\) with the imaginary octonions. The \(3\)-form determines a Riemannian metric \(g\) on \(M\) and hence a Hodge-star operator \(\star\). If \(\phi\) and \(\star\phi\) are closed, then the holonomy of \((M,g)\) is contained in \(G_2\), and if \(d\phi = \lambda\star\phi\) for some nonzero constant \(\lambda\) then \((M,g)\) is said to have weak holonomy \(G_2\).
A compact Lie group \(G\) is said to act with cohomogeneity one on a \(G_2\)-manifold \((M,\phi)\) if it preserves \(\phi\) and if there exists an orbit of codimension one. In this paper the authors determine all manifolds with holonomy or weak holonomy \(G_2\) that have a cohomogeneity one action of a connected compact simple Lie group \(G\). In this case \(G\) is either \(G_2\), \(Sp(2)\) or \(SU(3)\). Surprisingly, in each of the \(3\) cases there exists a unique solution with weak holonomy \(G_2\). For \(SU(3)\) they show that there is a one-parameter family of solutions with \(G_2\)-holonomy, whereas in the other two cases the solutions are isolated. As a special case they obtain the \(G_2\) manifolds found by Bryant and Salamon.
The approach by the authors is to consider a geodesic that is perpendicular to the principal orbits of the action, then to determine the solutions of the symplectic resp. cosymplectic \(G_2\) equation \(d\phi = 0\) resp. \(d\star\phi = 0\) along the geodesic, and finally to investigate the relevant topology and boundary conditions. The topological types of manifolds admitting a symplectic or cosymplectic \(G_2\)-structure are determined. They show that a symplectic \(G_2\)-structure exists only if \(M\) admits a holonomy \(G_2\)-metric, and construct new examples of compact cosymplectic \(G_2\)-structures.


53C29 Issues of holonomy in differential geometry
57M50 General geometric structures on low-dimensional manifolds
57S15 Compact Lie groups of differentiable transformations
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
Full Text: DOI arXiv


[1] L. Bérard Bergery, Sur de nouvelles variétés riemanniennes d’Einstein, Vol. 6, Publications de l’Institut É. Cartan, Nancy, 1982, pp. 1-60. · Zbl 0544.53038
[2] A.L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3, Folge, Vol. 10, Springer, Berlin, 1987. · Zbl 0613.53001
[3] A. Brandhuber, J. Gomis, S.S. Gubser, S. Gukov, Gauge theory at large N and new G2 holonomy metrics, June 2001. eprint arXiv:hep-th/0106034. · Zbl 0971.81071
[4] G.E. Bredon, Introduction to compact transformation groups, Pure Appl. Math. 46 (1972). · Zbl 0246.57017
[5] Bryant, R.L., Metrics with exceptional holonomy, Ann. math., 126, 525-576, (1987) · Zbl 0637.53042
[6] R.L. Bryant, Classical, exceptional, and exotic holonomies: a status report, Actes de la Table Ronde de Géométrie Différentielle, Luminy, 1992, Paris, Sémin. Congr., Vol. 1, Soc. Math. France, 1996, pp. 93-165. · Zbl 0882.53014
[7] Bryant, R.L.; Salamon, S.M., On the construction of some complete metrics with exceptional holonomy, Duke math. J., 58, 829-850, (1989) · Zbl 0681.53021
[8] Cabrera, F.M.; Monar, M.D.; Swann, A.F., Classification of G2-structures, J. London math. soc., 53, 407-416, (1996) · Zbl 0861.53024
[9] R. Cleyton, G-structures and Einstein metrics, Ph.D. Thesis, University of Southern Denmark, Odense, 2001. ftp://ftp.imada.sdu.dk/pub/phd/2001/24.PS.gz.
[10] M. Cvetič, G.W. Gibbons, H. Lü, C.N. Pope, New cohomogeneity-one metrics with Spin(7) holonomy, May 2001. eprint arXiv:math.DG/0105119.
[11] Friedrich, Th.; Kath, I.; Moroianu, A.; Semmelmann, U., On nearly parallel G2-structures, J. geom. phys., 23, 3-4, 259-286, (1997) · Zbl 0898.53038
[12] Glaeser, G., Fonctions composées différentiables, Ann. math., 77, 2, 193-209, (1963) · Zbl 0106.31302
[13] Gray, A., Weak holonomy groups, Math. Z., 123, 290-300, (1971) · Zbl 0222.53043
[14] N.J. Hitchin, Stable forms and special metrics, in: M.A. Fernández, J.A. Wolf (Eds.), Global Differential Geometry: The Mathematical Legacy of Alfred Gray, Am. Math. Soc., Providence, RI, 2001.
[15] Joyce, D., Compact Riemannian 7-manifolds with holonomy G2. I, J. diff. geom., 43, 291-328, (1996) · Zbl 0861.53022
[16] Joyce, D., Compact Riemannian 7-manifolds with holonomy G2. II, J. diff. geom., 43, 329-375, (1996) · Zbl 0861.53023
[17] D. Joyce, Compact Manifolds with Special Holonomy, Oxford Mathematical Monographs, Oxford University Press, 2000. · Zbl 1027.53052
[18] Kazdan, J.L.; Warner, F.W., Curvature functions for open 2-manifolds, Ann. math., 99, 2, 203-219, (1974) · Zbl 0278.53031
[19] A. Kovalev, Twisted connected sums and special Riemannian holonomy, Preprint MS-00-011, University of Edinburgh, December 2000. eprint arXiv:math.DG/0012189. · Zbl 1043.53041
[20] Mostert, P.S.; Mostert, P.S., On a compact Lie group acting on a manifold, Ann. math., Ann. math., 66, 589-455, (1957) · Zbl 0080.16702
[21] Swann, A.F., Hyper-Kähler and quaternionic Kähler geometry, Math. ann., 289, 421-450, (1991) · Zbl 0711.53051
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.