SU(3)-manifolds of cohomogeneity one. (English) Zbl 1155.57033
A cohomogeneity-one \(G\)-space \(M\) is a smooth manifold with a smooth action of a compact semisimple group such that the principal orbits are codimension-one submanifolds. Such a \(G\)-space has the following nice property (as shown by [P. S. Mostert, Ann. Math. (2) 65, 447–455 (1957); errata ibid. 66, 589 (1957; Zbl 0080.16702)]) that if \(M\) is compact, then the orbit space \(M/G\) is isomorphic to the interval \([0,1]\) or \(S^1\); if \(M\) is non-compact, then \(M/G\) is isomorphic to \([0,1)\) or \({\mathbb R}\). In particular, in the case where \(M/G=[0,1]\), there are precisely two singular orbits corresponding to the endpoints of \([0,1]\). The study on such \(G\)-spaces has had an increasing development in the recent years. For example, [A. Dancer and A. Swann, J. Geom. Phys. 21, No. 3, 218–230 (1997; Zbl 0909.53032) and Int. J. Math. 10, No. 5, 541-570 (1999; Zbl 1066.53510)] classified cohomogeneity-one hyper-Kähler and compact quaternion-Kähler \(G\)-manifolds if \(G\) is simple or semisimple.
The paper of under review considers 7- and 8-dimensional smooth manifolds \(M\) with an action of \(SU(3)\) of cohomogeneity-one, and gives the classification result in the following cases (1) \(M\) is simply connected and \(M/SU(3)=[0,1]\), and (2) \(M/G=S^1\) and the principal orbits are simply connected. In addition, the result is also applied to the study of \(SU(3)\) as a manifold, the 8-dimensional quaternion-Kähler manifolds etc.
The paper of under review considers 7- and 8-dimensional smooth manifolds \(M\) with an action of \(SU(3)\) of cohomogeneity-one, and gives the classification result in the following cases (1) \(M\) is simply connected and \(M/SU(3)=[0,1]\), and (2) \(M/G=S^1\) and the principal orbits are simply connected. In addition, the result is also applied to the study of \(SU(3)\) as a manifold, the 8-dimensional quaternion-Kähler manifolds etc.
Reviewer: Zhi Lü (Shanghai)
MSC:
| 57S25 | Groups acting on specific manifolds |
| 57S15 | Compact Lie groups of differentiable transformations |
| 22E46 | Semisimple Lie groups and their representations |
| 53C30 | Differential geometry of homogeneous manifolds |
| 53C26 | Hyper-Kähler and quaternionic Kähler geometry, “special” geometry |
| 58D05 | Groups of diffeomorphisms and homeomorphisms as manifolds |
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