On the topology of G-manifolds with finitely many non-principal orbits

Bechtluft-Sachs, Stefan; Wraith, David J.

doi:10.1016/j.topol.2012.07.008

Cited by 2 publications

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References 23 publications

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“…Indeed it might be possible that no manifold with more than two singular orbits can support an invariant metric with positive Ricci curvature. 4. Are there any simply-connected examples which do not admit an invariant metric with positive Ricci curvature?…”

Section: Open Problemsmentioning

confidence: 99%

“…Thus, for instance, there are actions of U(1) on S 2k+1 and of SU(2) on S 2k with only one non-principal orbit. We will show that many of the examples of G-manifolds with finitely many non-principal orbits constructed in [4] admit an invariant metric with positive Ricci curvature. We recall a few examples: Example 2 ('Doubles') Let L be finite or one of U(1), SU (2), N SU(2) U(1).…”

Section: Introductionmentioning

confidence: 99%

“…By Theorem 6.2 in [5] the group L i must finite or one of the groups U(1), SU (2), N SU(2) U(1). For details on this and the subsequent topological facts about G-manifolds with finitely many non-principal orbits we refer to [4]. From now on we will assume that the cohomogeneity is at least 2.…”

Section: Introductionmentioning

confidence: 99%

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On the curvature on G-manifolds with finitely many non-principal orbits

Bechtluft-Sachs

Wraith

2012

Geom Dedicata

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Section: Open Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the curvature on G-manifolds with finitely many non-principal orbits

Bechtluft-Sachs

Wraith

2012

Geom Dedicata

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“…It turns out that such actions are quite tightly constrained (see [B] chapter 4 §6), and this results in the following Proposition 1.1. ([BW1], Theorem 9). If the cohomogeneity is greater than one, then K is ineffective kernel of the H i action on S r , so K is normal in H i and H i /K ∼ = U (1), N SU(2) U (1), SU (2), or is finite, and acts freely and linearly on the normal sphere S r .…”

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confidence: 98%

“…Using the symbol × α to denote a quotient under this action, we have Proposition 1.4. ( [BW1], Theorem 3). For a small invariant tubular neighbourhood N of a singular orbit G/H, we have N ∼ = D r+1 × α G/K.…”

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$$G$$ G -manifolds with positive Ricci curvature and many isolated singular orbits

Wraith

2013

Ann Glob Anal Geom

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We show that in cohomogeneity 3 there are G-manifolds with any given number of isolated singular orbits and an invariant metric of positive Ricci curvature. We show that the corresponding result is also true in cohomogeneity 5 provided the number of singular orbits is even. §1 IntroductionThe objects under consideration in this paper are compact G-manifolds with finitely many non-principal orbits. Here, G is a compact Lie group acting smoothly and effectively on a smooth compact manifold M. The orbits of such an action are either principal, exceptional (that is, non-principal but having the same dimension as a principal orbit), or singular, meaning that the orbit dimension is strictly lower than that of a principal orbit. The cohomogeneity of such an action is the codimension of a principal orbit, or equivalently the dimension of the space of orbits G\M. Note that the union of principal orbits is a dense subset of M.We will be primarily interested in the invariant geometry of such objects, that is, the geometry of M equipped with a Riemannian metric which is invariant under the G-action.The motivation for studying manifolds with only finitely many non-principal orbits arose from the study of cohomogeneity one manifolds. Cohomogeneity one manifolds have been studied extensively in recent years, particularly for their geometric properties. (See for example the survey [Z].) Recently, a new example of a positively curved manifold was found among the cohomogeneity one manifolds ([D],[GVZ]), to add to the many known examples of non-negatively curved cohomogeneity-one manifolds (see for example [GZ1]). The underlying philosophy behind these developments is that curvature or other geometric expressions become simpler and more tractable in the prescence of symmetry, as provided by the action of a 'large' Lie group acting isometrically.Compact cohomogeneity one manifolds belong to one of two types according to the space of orbits. The orbit space could be a circle, in which case all orbits are principal and the manifold is a principal orbit bundle over the circle. The other possiblilty is a closed interval, in which case there are precisely two non-principal orbits corresponding to the end-points of the interval. This is by far the more interesting of the two cases, and provides our main motivating example for studying manifolds with a finite number of non-principal orbits. Such orbits are clearly isolated in the sense that there exist mutually disjoint invariant tubular neighbourhoods about each non-principal orbit.2000 Mathematics Subject Classification: 53C20, 53C21.

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On the topology of G-manifolds with finitely many non-principal orbits

Cited by 2 publications

References 23 publications

On the curvature on G-manifolds with finitely many non-principal orbits

On the curvature on G-manifolds with finitely many non-principal orbits

$$G$$ G -manifolds with positive Ricci curvature and many isolated singular orbits

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