Tohoku Mathematical Journal

Polar foliations on quaternionic projective spaces

Miguel Domínguez-Vázquez and Claudio Gorodski

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We classify irreducible polar foliations of codimension $q$ on quaternionic projective spaces $\mathbb{H} P^n$, for all $(n,q)\neq(7,1)$. We prove that all irreducible polar foliations of any codimension (resp. of codimension one) on $\mathbb{H} P^n$ are homogeneous if and only if $n+1$ is a prime number (resp. $n$ is even or $n=1$). This shows the existence of inhomogeneous examples of codimension one and higher.

Article information

Source
Tohoku Math. J. (2), Volume 70, Number 3 (2018), 353-375.

Dates
First available in Project Euclid: 21 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1537495351

Digital Object Identifier
doi:10.2748/tmj/1537495351

Mathematical Reviews number (MathSciNet)
MR3856771

Zentralblatt MATH identifier
06996532

Subjects
Primary: 53C12: Foliations (differential geometric aspects) [See also 57R30, 57R32]
Secondary: 53C35: Symmetric spaces [See also 32M15, 57T15] 57S15: Compact Lie groups of differentiable transformations

Keywords
polar foliation singular Riemannian foliation $s$-representation symmetric space FKM-foliation homogeneous foliation quaternionic projective space

Citation

Domínguez-Vázquez, Miguel; Gorodski, Claudio. Polar foliations on quaternionic projective spaces. Tohoku Math. J. (2) 70 (2018), no. 3, 353--375. doi:10.2748/tmj/1537495351. https://projecteuclid.org/euclid.tmj/1537495351


Export citation

References

  • U. Abresch, Isoparametric hypersurfaces with four or six distinct principal curvatures, Math. Ann. 264 (1983), 283–302.
  • M. M. Alexandrino, Singular Riemannian foliations with sections, Illinois J. Math. 48 (2004), 1163–1182.
  • I. Bergmann, Reducible polar representations, Manuscripta Math. 104 (2001), 309–324.
  • A. L. Besse, Manifolds all of whose geodesics are closed}, Ergebnisse der Mathematik und ihrer Grenzgebiete, 93}, Springer-Verlag, Berlin-New York, 1978.
  • A. L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 10, Springer-Verlag, Berlin, 1987.
  • É. Cartan, Sur quelques familles remarquables d'hypersurfaces, C. R. Congrès Math. Liège (1939), 30–41.
  • T. Cecil, Q.-S. Chi and G. Jensen, Isoparametric hypersurfaces with four principal curvatures, Ann. of Math. (2) 166 (2007), 1–76.
  • Q.-S. Chi, Isoparametric hypersurfaces with four principal curvatures, III, J. Differential Geom. 94 (2013), no. 3, 469–504.
  • U. Christ, Homogeneity of equifocal submanifolds, J. Differential Geom. 62 (2002), no. 1, 1–15.
  • D. H. Collingwood and W. M. McGovern, Nilpotent orbits in semisimple Lie algebras}, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993.
  • J. Dadok, Polar coordinates induced by actions of compact Lie groups, Trans. Amer. Math. Soc. 288 (1985), no. 1, 125–137.
  • M. Domínguez-Vázquez, Isoparametric foliations on complex projective spaces, Trans. Amer. Math. Soc. 368 (2016), no. 2, 1211–1249.
  • J. Dorfmeister and E. Neher, Isoparametric hypersurfaces, case $g=6$, $m=1$, Commun. Algebra 13 (1985), 2299–2368.
  • J.-H. Eschenburg and E. Heintze, On the classification of polar representations, Math. Z. 232 (1999), 391–398.
  • F. Fang, K. Grove and G. Thorbergsson, Tits geometry and positive curvature, Acta Math. 218 (2017), no. 1, 1–53. arXiv:1205.6222v2 [math.DG]
  • D. Ferus, H. Karcher and H. F. Münzner, Cliffordalgebren und neue isoparametrische Hyperflächen, Math. Z. 177 (1981), no. 4, 479–502.
  • T. Foertsch and A. Lytchak, The de Rham decomposition theorem for metric spaces, Geom. Funct. Anal. 18 (2008), no. 1, 120–143.
  • J. Ge, Z. Tang and W. Yan, A filtration for isoparametric hypersurfaces in Riemannian manifolds, J. Math. Soc. Japan 67 (2015), no. 3, 1179–1212.
  • C. Gorodski and E. Heintze, Homogeneous structures and rigidity of isoparametric submanifolds in Hilbert space, J. Fixed Point Theory Appl. 11 (2012), 93–136.
  • C. Gorodski and A. Lytchak, On orbit spaces of representations of compact Lie groups, J. Reine Angew. Math. 691 (2014), 61–100.
  • C. Gorodski and A. Lytchak, Isometric actions on spheres with an orbifold quotient, Math. Ann. 365 (2016), no. 3–4, 1041–1067.
  • E. Heintze, X. Liu and C. Olmos, Isoparametric submanifolds and a Chevalley-type restriction theorem, Integrable systems, geometry, and topology, 151–190, AMS/IP Stud. Adv. Math., 36, Amer. Math. Soc., Providence, RI, 2006.
  • S. Helgason, Differential geometry, Lie groups, and symmetric spaces}, Pure and Applied Mathematics, 80, Academic Press, Inc., New York-London, 1978.
  • S. Immervoll, On the classification of isoparametric hypersurfaces with four principal curvatures in spheres, Ann. of Math. (2) 168 (2008), 1011–1024.
  • A. Lytchak, Polar foliations of symmetric spaces, Geom. Funct. Anal. 24 (2014), no. 4, 1298–1315.
  • R. Miyaoka, Isoparametric hypersurfaces with $(g,m)=(6,2)$, Ann. of Math. (2) 177 (2013), no. 1, 53–110.
  • R. Miyaoka, Errata of “Isoparametric hypersurfaces with $(g,m) = (6,2)$”, Ann. of Math. (2) 183 (2016), no. 3, 1057–1071.
  • H. F. Münzner, Isoparametrische Hyperflächen in Sphären, Math. Ann. 251 (1980), no. 1, 57–71.
  • H. F. Münzner, Isoparametrische Hyperflächen in Sphären II. Über die Zerlegung der Sphäre in Ballbündel, Math. Ann. 256 (1981), no. 2, 215–232.
  • F. Podestà and G. Thorbergsson, Polar actions on rank-one symmetric spaces, J. Differential Geom. 53 (1999), 131–175.
  • M. Radeschi, Low dimensional singular Riemannian foliations in spheres, arXiv:1203.6113 [math.DG].
  • A. Siffert, A new structural approach to isoparametric hypersurfaces in spheres, Ann. Global Anal. Geom. 52 (2017), no. 4, 425–456.
  • G. Thorbergsson, Isoparametric foliations and their buildings, Ann. of Math. (2) 133 (1991), 429–446.
  • G. Thorbergsson, Singular Riemannian foliations and isoparametric submanifolds, Milan J. Math. 78 (2010), no. 1, 355–370.
  • D. A. Vogan Jr., Three-dimensional subgroups and unitary representations, Challenges for the 21st century (Singapore, 2000), 213–250, World Sci. Publ., River Edge, NJ, 2001.
  • J. A. Wolf, Spaces of constant curvature. Sixth edition}, AMS Chelsea Publishing, Providence, RI, 2011.
  • L. Xiao, Principal curvatures of isoparametric hypersurfaces in $\mathbb{C} P^n$, Trans. Amer. Math. Soc. 352 (2000), no. 10, 4487–4499.