Abstract
An isometricH-action on a Riemannian manifoldX is calledpolar if there exists a closed submanifoldS ofX that meets everyH-orbit and always meets orbits orthogonally (S is called a section). LetG be a compact Lie group equipped with a biinvariant metric,H a closed subgroup ofG ×G, and letH act onG isometrically by (h 1,h 2) ·x = h 1 xh −12 · LetP(G, H) denote the group ofH 1-pathsg: [0, 1] →G such that (g(0),g (1)) ∈H, and letP(G, H) act on the Hilbert spaceV = H 0([0, 1], g) isometrically byg * u = gug −1 −g′g −1. We prove that if the action ofH onG is polar with a flat section then the action ofP(G, H) onV is polar. Principal orbits of polar actions onV are isoparametric submanifolds ofV and are infinite-dimensional generalized real or complex flag manifolds. We also note that the adjoint actions of affine Kac-Moody groups and the isotropy action corresponding to an involution of an affine Kac-Moody group are special examples ofP(G, H)-actions for suitable choice ofH andG.
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References
Bott, R., and Samelson, H., Applications of the theory of Morse to symmetric spaces.Amer. J. Math. 80, 964–1029 (1958).
Bourbaki, N.,Groupes et Algebres de Lie. Hermann, Paris, 1968.
Conlon, L., The topology of certain spaces of paths on a ocmpact symmetric space.Trans. Amer. Math. Soc. 112, 228–248 (1964).
Conlon, L., Variational completeness and K-transversal domains.J. Differential Geom. 5, 135–147 (1971).
Conlon, L., A class of variationally complete representations.J. Differential Geom. 7, 149–160 (1972).
Dadok, J., Polar coordinates induced by actions of compact Lie groups.Trans. Amer. Math. Soc. 288, 125–137 (1985).
Ferus, D., Karcher, H., and Münzner, H. F., Cliffordalgebren und neue isoparametrische hyperflächen.Math. Z. 177, 479–502 (1981).
Helgason, S.,Differential Geometry and Symmetric Spaces. Academic Press, 1978.
Hermann, R., Variational completeness for compact symmetric spaces.Proc. Amer. Math. Soc. 11, 554–546 (1960).
Kac, V. G.,Infinite Dimensional Lie Algebras. Cambridge University Press, 1985.
Ozeki, H., and Takeuchi, M., On some types of isoparametric hypersurfaces in spheres, I.Tohoku Math. J. 127, 515–559 (1975).
Ozeki, H., and Takeuchi, M., On some types of isoparametric hypersurfaces in spheres, II.Tohoku Math. J. 28, 7–55 (1976).
Palais, R. S., Morse theory on Hilbert manifolds.Topology 2, 299–340 (1963).
Palais, R. S.,Foundations of Global Non-linear Analysis. Benjamin Co., New York, 1968.
Palais, R. S., and Terng, C. L., A general theory of canonical forms.Trans. Amer. Math. Soc. 300, 771–789 (1987).
Palais, R. S., and Terng, C. L.,Critical Point Theory and Submanifold Geometry, Lecture Notes in Math., vol. 1353. Springer-Verlag, Berlin and New York, 1988.
Pinkall, U., and Thorbergsson, G., Examples of infinite dimensional isoparametric submanifolds.Math. Z. 205, 279–286 (1990).
Pressley, A., and Segal, G. B.,Loop Groups. Oxford Science Publ., Clarendon Press, Oxford, 1986.
Terng, C. L., Isoparametric submanifolds and their Coxeter groups.J. Differential Geom. 21, 79–107 (1985).
Terng, C. L., Proper Fredholm submanifolds of Hilbert spaces.J. Differential Geom. 29, 9–47 (1989).
Terng, C. L., Variational completeness and infinite dimensional geometry.Proceedings of Leeds Conference, edited by L. Verstraelen and A. West.Geometry and Topology of Submanifolds, III, pp. 279–293. World Scientific, Singapore, 1991.
Thorbergsson, G., Isoparametric foliations and their buildings.Annals of Math. 133, 429–446 (1991).
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Work supported partially by NSF Grant DMS 8903237 and by The Max-Planck-Institut für Mathematik in Bonn.
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Terng, CL. Polar actions on Hilbert space. J Geom Anal 5, 129–150 (1995). https://doi.org/10.1007/BF02926445
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DOI: https://doi.org/10.1007/BF02926445