On the topology of \(G\)-manifolds with finitely many non-principal orbits. (English) Zbl 1255.57029
In this paper, for a compact Lie-group \(G\), smooth \(G\)-manifolds are studied. The authors give a classification of those compact \(G\)-manifolds which have only finitely many non-principal orbits. This class of manifolds is a natural generalization of cohomogeneity-one manifolds which have zero or two non-principal orbits.
Using their classification the authors construct infinite families of \(11\)-dimensional and \(13\)-dimensional \(SU(3)\)-manifolds with two non-principal orbits which are equal to Aloff-Wallach spaces. They also show that these families contain infinitely many pairwise non-homotopy equivalent examples.
Moreover, the question is studied how many non-principal orbits a \(G\)-manifold with finitely many non-principal orbits can have.
Using their classification the authors construct infinite families of \(11\)-dimensional and \(13\)-dimensional \(SU(3)\)-manifolds with two non-principal orbits which are equal to Aloff-Wallach spaces. They also show that these families contain infinitely many pairwise non-homotopy equivalent examples.
Moreover, the question is studied how many non-principal orbits a \(G\)-manifold with finitely many non-principal orbits can have.
Reviewer: Michael Wiemeler (Karlsruhe)
MSC:
| 57S15 | Compact Lie groups of differentiable transformations |
| 53C20 | Global Riemannian geometry, including pinching |
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