Compact connected Lie transformation groups on spheres with low cohomogeneity. I

Straume, Eldar

doi:10.1090/memo/0569

Cited by 31 publications

(34 citation statements)

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“…In this section we will list all isometric cohomogeneity one actions on compact simply connected symmetric spaces of dimension seven or less. Hsiang and Lawson [1971] classified cohomogeneity one actions on symmetric spheres in (see [Straume 1996] for correction) and later Uchida [1977] did the same for complex projective spaces. Kollross [2002] generalized these results to a classification of cohomogeneity one actions on irreducible symmetric spaces of compact type.…”

Section: Identifying Some Actionsmentioning

confidence: 97%

Classification of cohomogeneity one manifolds in low dimensions

Hoelscher¹

2010

Pacific J. Math.

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A cohomogeneity one manifold is a manifold whose quotient by the action of a compact Lie group is one-dimensional. Such manifolds are of interest in Riemannian geometry in the context of nonnegative sectional curvature, as well as in other areas of geometry and in physics. We classify compact simply connected cohomogeneity one manifolds in dimensions 5, 6 and 7. We also show that all such manifolds admit metrics of nonnegative sectional curvature, with the possible exception of two families of manifolds.

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Section: Identifying Some Actionsmentioning

confidence: 97%

Classification of cohomogeneity one manifolds in low dimensions

Hoelscher¹

2010

Pacific J. Math.

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“…According to [26], aside from cohomogeneity one actions on the standard spheres, only exotic Kervaire spheres admit cohomogeneity one actions. More precisely, every exotic Kervaire sphere can be represented as a smooth submanifold Σ 2n−1 d of C n+1 defined by the equations…”

Section: Remark 33mentioning

confidence: 99%

Isoparametric functions on exotic spheres

Qian

Tang

2015

Advances in Mathematics

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This paper extends widely the work in [11]. Existence and non-existence results of isoparametric functions on exotic spheres and Eells-Kuiper projective planes are established. In particular, every homotopy n-sphere (n > 4) carries an isoparametric function (with certain metric) with 2 points as the focal set, in strong contrast to the classification of cohomogeneity one actions on homotopy spheres [26] (only exotic Kervaire spheres admit cohomogeneity one actions besides the standard spheres). As an application, we improve a beautiful result of Bérard-Bergery [2] (see also pp. 234-235 of [3]).

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“…The orthogonal action of SOn on S n is an example of the special P-manifolds de®ned by Ja Ènich [13, 1.2]. Other examples include the`cohomogeneity 1' actions studied by E. Straume [25] (see [7] for a recent application and other references). In this section we give the classi®cation of equivariant P; G-bundles over special P-manifolds.…”

Section: A More General Settingmentioning

confidence: 99%