Journal of the Mathematical Society of Japan

A filtration for isoparametric hypersurfaces in Riemannian manifolds

Jianquan GE, Zizhou TANG, and Wenjiao YAN

Full-text: Open access

Abstract

This paper introduces the notion of $k$-isoparametric hypersurface in an $(n+1)$-dimensional Riemannian manifold for $k=0,1,\dots,n$. Many fundamental and interesting results (towards the classification of homogeneous hypersurfaces among other things) are given in complex projective spaces, complex hyperbolic spaces, and even in locally rank one symmetric spaces.

Article information

Source
J. Math. Soc. Japan, Volume 67, Number 3 (2015), 1179-1212.

Dates
First available in Project Euclid: 5 August 2015

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1438777446

Digital Object Identifier
doi:10.2969/jmsj/06731179

Mathematical Reviews number (MathSciNet)
MR3376584

Zentralblatt MATH identifier
1331.53086

Subjects
Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42] 53C24: Rigidity results

Keywords
isoparametric hypersurface constant mean curvature rank one symmetric space Riccati equation Chern conjecture

Citation

GE, Jianquan; TANG, Zizhou; YAN, Wenjiao. A filtration for isoparametric hypersurfaces in Riemannian manifolds. J. Math. Soc. Japan 67 (2015), no. 3, 1179--1212. doi:10.2969/jmsj/06731179. https://projecteuclid.org/euclid.jmsj/1438777446


Export citation

References

  • U. Abresch, Isoparametric hypersurfaces with four or six distinct principal curvatures, Math. Ann., 264 (1983), 283–302.
  • J. Berndt, Real hypersurfaces with constant principal curvatures in complex hyperbolic space, J. Reine Angew. Math., 395 (1989), 132–141.
  • J. Berndt, Real hypersurfaces in quaternionic space forms, J. Reine Angew. Math., 419 (1991), 9–26.
  • J. Berndt, A note on hypersurfaces in symmetric spaces, Proceedings of the Fourteenth International Workshop on Diff. Geom., 14 (2010), 1–11.
  • M. Brozos-Vázquez, P. Gilkey, and S. Nikčević, Geometric realizations of curvature, Nihonkai Math. J., 20 (2009), 1–24.
  • J. Berndt and H. Tamaru, Cohomogeneity one actions on noncompact symmetric spaces of rank one, Trans. Amer. Math. Soc., 359 (2007), 3425–3438.
  • E. Cartan, Familles de surfaces isoparamétriques dans les espaces à courbure constante, Annali di Mat., 17 (1938), 177–191.
  • E. Cartan, Sur des familles remarquables d'hypersurfaces isoparamétriques dans les espaces sphériques, Math. Z., 45 (1939), 335–367.
  • E. Cartan, Sur quelque familles remarquables d'hypersurfaces, In: C. R. Congrès Math. Liège, 1939, pp.,30–41.
  • E. Cartan, Sur des familles d'hypersurfaces isoparamétriques des espaces sphériques à 5 et à 9 dimensions, Revista Univ. Tucuman, Serie A, 1 (1940), 5–22.
  • T. E. Cecil, Q. S. Chi and G. R. Jensen, Isoparametric hypersurfaces with four principal curvatures, Ann. Math., 166 (2007), 1–76.
  • T. E. Cecil, Isoparametric and Dupin Hypersurfaces, SIGMA, 4 (2008), arXiv:0809.1433.
  • P. Carpenter, A. Gray and T. J. Willmore, The curvature of Einstein symmetric spaces, Quart. J. Math. Oxford, 33 (1982), 45–64.
  • Q. S. Chi, A curvature characterization of certain locally rank one symmetric spaces, J. Diff. Geom., 28 (1988), 187–202.
  • Q. S. Chi, Isoparametric hypersurfaces with four principal curvatures, II, Nagoya Math. J., 204 (2011), 1–18.
  • T. E. Cecil and P. T. Ryan, Tight and taut immersions of manifolds, Research Notes in Math., 107, Pitman, London, 1985.
  • J. C. Díaz-Ramos and M. Domínguez-Vázquez, Inhomogeneous isoparametric hypersurfaces in complex hyperbolic spaces, Math. Z., 271 (2012), 1037–1042.
  • J. Dorfmeister and E. Neher, Isoparametric hypersurfaces, case $g = 6$, $m = 1$, Comm. Algebra, 13 (1985), 2299–2368.
  • D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Springer-Verlag, New York, 1995.
  • D. Ferus, H. Karcher, and H. F. Münzner, Cliffordalgebren und neue isoparametrische Hyperflächen, Math. Z., 177 (1981), 479–502.
  • K. Grove and S. Halperin, Dupin hypersurfaces, group actions, and the double mapping cylinder, J. Diff. Geom., 26 (1987), 429–459.
  • A. Gray, Tubes, Second Edition, Progr. Math., 221, Birkhäuser Verlag Basel–Boston–Berlin, 2004.
  • J. Q. Ge and Z. Z. Tang, Chern conjecture and isoparametric hypersurfaces, In: Differential Geometry: Under the Influence of S.-S. Chern, the Advanced Lectures in Mathematics series, 22, (edi. Y. B. Shen, Z. M. Shen and S. T. Yau), International Press, 2012, pp.,49–60.
  • J. Q. Ge and Z. Z. Tang, Isoparametric functions and exotic spheres, J. Reine Angew. Math., 683 (2013), 161–180.
  • J. Q. Ge and Z. Z. Tang, Geometry of isoparametric hypersurfaces in Riemannian manifolds, Asian J. Math., 18 (2014), 117–126.
  • 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. Immervoll, On the classification of isoparametric hypersurfaces with four distinct principal curvatures in spheres, Ann. Math., 168 (2008), 1011–1024.
  • M. Kimura, Real hypersurfaces and complex submanifolds in complex projective space, Trans. Amer. Math. Soc., 296 (1986), 137–149.
  • H. Matsumura, Commutative Algebra, Second Edition, Mathematics Lecture Note Series, 56, Benjamin/Cummings Publishing Company, Inc., London, Amsterdam, 1980.
  • R. Miyaoka, Isoparametric hypersurfaces with $(g,m)=(6,2)$, Ann. of Math., 177 (2013), 53–110.
  • R. Miyaoka, Transnormal functions on a Riemannian manifold, Diff. Geom. Appl., 31 (2013), 130–139.
  • H. F. Münzner, Isoparametric hyperflächen in sphären, I and II, Math. Ann., 251 (1980), 57–71 and 256 (1981), 215–232.
  • T. Murphy, Real hypersurfaces in complex and quaternionic space forms, Adv. Geom., 14 (2014), 1–10.
  • Y. Nikolayevsky, Osserman Conjecture in dimension $n \neq 8$, $16$, Math. Ann., 331 (2005), 505–522.
  • H. Ozeki and M. Takeuchi, On some types of isoparametric hypersurfaces in spheres, I and II, Tohoku Math. J., 27 (1975), 515–559 and 28 (1976), 7–55.
  • K. S. Park, Isoparametric families on projective spaces, Math. Ann., 284 (1989), 503–513.
  • C. Qian and Z. Z. Tang, Recent progress in isoparametric functions and isoparametric hypersurfaces, Real and Complex Submanifolds, Springer Proceedings in Mathematics & Statistics, 106 (2014), 65–76.
  • C. Qian and Z. Z. Tang, Isoparametric functions on exotic spheres, Adv. Math., 272 (2015), 611–629.
  • L. Smith, Polynomial invariants of finite groups, A. K. Peters, Wellesley, MA, 1995.
  • M. Scherfner and S. Weiss, Towards a proof of the Chern conjecture for isoparametric hypersurfaces in spheres,
  • Proc., South German Diff. Geom. Colloq., Vienna, 2008, pp.,1–13.
  • R. Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math., 10 (1973), 495–506.
  • R. Takagi, A class of hypersurfaces with constant principal curvatures in a sphere, J. Diff. Geom., 11 (1976), 225–233.
  • Z. Z. Tang, Multiplicities of equifocal hypersurfaces in symmetric spaces, Asian J. Math., 2 (1998), 181–213.
  • G. Thorbergsson, A survey on isoparametric hypersurfaces and their generalizations, In Handbook of Diff. Geom., I, North-Holland, Amsterdam, 2000, 963–995.
  • C. L. Terng and G. Thorbergsson, Submanifold geometry in symmetric spaces, J. Diff. Geom., 42 (1995), 665–718.
  • Z. Z. Tang, Y. Q. Xie and W. J. Yan, Isoparametric foliation and Yau conjecture on the first eigenvalue, II, J. Funct. Anal., 266 (2014), 6174–6199.
  • Z. Z. Tang and W. J. Yan, Isoparametric foliation and Yau conjecture on the first eigenvalue, J. Diff. Geom., 94 (2013), 521-540.
  • Q. M. Wang, Isoparametric hypersurfaces in complex projective spaces, Diff. Geom. and Diff. Equ., Proc., 1980 Beijing Sympos., 3, 1982, 1509–1523.
  • Q. M. Wang, Isoparametric Functions on Riemannian Manifolds. I, Math. Ann., 277 (1987), 639–646.
  • L. Xiao, Principal curvatures of isoparametric hypersurfaces in $\mathbb{C}P^n$, Proc. Amer. Math. Soc., 352 (2000), 4487–4499.