From isoparametric submanifolds to polar foliations

Thorbergsson, Gudlaugur

doi:10.1007/s40863-022-00291-2

Cited by 4 publications

(2 citation statements)

References 35 publications

(72 reference statements)

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“…Further, in view of (3.3) these hyperbolic planes are totally geodesic if and only if β = 0 if and only if the foliation by integral curves of X is Riemannian (indeed a polar foliation, see e.g. [Thr22]) if and only if the projection of SL(2, R) onto the space of leaves is a Riemannian submersion; equivalently, Y (F ) = 0. Assume this is the case; we work on the open set U defined by F ̸ = 0, where we can take a new frame T = T , X = F −1 X, Ỹ = Y .…”

Section: κ-Nullitymentioning

confidence: 99%

Generalized warped products and the κ-nullity of Riemannian curvature

Gorodski¹,

Guimarães²

2022

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Section: κ-Nullitymentioning

confidence: 99%

Generalized warped products and the κ-nullity of Riemannian curvature

Gorodski¹,

Guimarães²

2022

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“…Much of the theory of polar actions has been generalized to "singular Riemannian sections with sections", or "polar foliations" as they are now called, but the lack of group action causes some difficulties. We will not discuss them here and instead refer to [AB15,Rad17,Tho22] for discussions and references. 1 Lecture 1: Polar actions…”

Section: Introductionmentioning

confidence: 99%

Topics in polar actions

Gorodski¹

2022

Preprint

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We wish to thank Uwe Semmelmann and Andreas Kollross for the invitation to give these lectures. We assume basic knowledge of isometric actions on Riemannian manifolds, including the normal slice theorem and the principal orbit type theorem. Lecture 1 introduces polar actions and culminates with Heintze, Liu and Olmos's argument to characterize them in terms of integrability of the distribution of normal spaces to the principal orbits. The other two lectures are devoted to two of Lytchak and Thorbergsson's results. In Lecture 2 we briefly review Riemannian orbifolds from the metric point of view, and explain their characterization of orbifold points in the orbit space of a proper and isometric action in terms of polarity of the slice representation above. In Lecture 3 we present their proof of the fact that variationally complete actions in the sense of Bott and Samelson on non-negatively curved manifolds are hyperpolar. The appendix contains explanations of some results used in the lectures, namely: a criterion for the polarity of isometric actions on symmetric spaces, a discussion of Cartan's and Hermann's criterions for the existence of totally geodesic submanifolds, and a more or less self-contained derivation of Wilking's transversal Jacobi equation.

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