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Polar Foliations

  • Chapter
Lie Groups and Geometric Aspects of Isometric Actions

Abstract

In this chapter, we give a survey on some results from the theory of polar foliations, also called singular Riemannian foliations with sections. This theory is a generalization of the classical theory of adjoint actions (presented in Chap. 4), and several results in this chapter are extensions of its results. Since our main goal is to provide a flavor of this new field, to not lose sight of the big picture, we only give some sketches of proofs without going into technical details.

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Notes

  1. 1.

    The linear connection on O(T Σ) is defined as follows. For a fixed frame ξ p , set \(\widehat{\mathcal{H}}_{\xi _{p}}\) as the space of all vectors \(\widehat{\alpha }'(0)\), where \(\widehat{\alpha }\) is the parallel transport of ξ p along a curve \(\alpha: [0, 1] \rightarrow \varSigma\) with α(0) = p. 

  2. 2.

    This is possible since the set of regular leaves is open and dense in M.

  3. 3.

    Alternatively, one can prove that dη t ξ has constant rank for t close to 1 using ideas presented in the proof of Theorem 5.63.

  4. 4.

    The metric \(\mathtt{g}_{0}\) is such that \(\exp _{x}\) becomes an isometry.

  5. 5.

    I.e., by composing the inverse of the normal exponential map \(\exp _{x}: B_{\epsilon }(0) \cap T_{x}S_{x} \rightarrow S_{x}\) with an isometry between \(T_{x}S_{x}\) and \(\mathbb{R}^{n}\).

  6. 6.

    I.e., a quotient of a simply-connected Riemannian manifold by a reflection group in the classical sense , i.e., a discrete group of isometries generated by reflections acting properly and effectively.

  7. 7.

    I.e., for any two points p and q in Ω, every minimal geodesic segment between p and q lies entirely in Ω.

  8. 8.

    For details on compact rank one symmetric spaces, see Example 6.10 .

  9. 9.

    According to Thorbergsson [209], the term isoparametric hypersurface was introduced by Levi-Civita and \(\|\mathrm{grad}\ f\|^{2}\) and Δ f were called the differential parameters of f. We stress that, in modern references, the term isoparametric hypersurface is used for level sets of isoparametric functions, see Remark 5.58.

  10. 10.

    Here, a point of a section is called regular if there exists only one local section that contains p, i.e., if two local sections σ and \(\widetilde{\sigma }\) contain p, then they have the same germ at p.

  11. 11.

    Here, a basic function is a function constant along the leaves.

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Authors and Affiliations

  1. Departamento de Matemática, Instituto de Matemática e Estatística Universidade de São Paulo, São Paulo, Brazil

    Marcos M. Alexandrino

  2. Department of Mathematics, University of Pennsylvania, Philadelphia, PA, USA

    Renato G. Bettiol

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  1. Marcos M. Alexandrino
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  2. Renato G. Bettiol
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Alexandrino, M.M., Bettiol, R.G. (2015). Polar Foliations. In: Lie Groups and Geometric Aspects of Isometric Actions. Springer, Cham. https://doi.org/10.1007/978-3-319-16613-1_5

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