Top

Published in:

2016 | OriginalPaper | Chapter

Differential Topology Interacts with Isoparametric Foliations

Authors : Chao Qian, Jianquan Ge

Published in: Geometry and Topology of Manifolds

Publisher: Springer Japan

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

In this note, we discuss the interactions between differential topology and isoparametric foliations, surveying some recent progress and open problems.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

previous chapter Can One Hear the Shape of a Group?

next chapter Unobstructed Deformations of Generalized Complex Structures Induced by Logarithmic Symplectic Structures and Logarithmic Poisson Structures

The supscript “+” means that diffeomorphisms are orientation-preserving.

Here by “exotic” we mean non-equivalent as disk bundles but with diffeomorphic total spaces.

Akhmedov, A., Doug, B.: Park, exotic smooth structures on small \(4\)-manifolds with odd signatures. Invent. Math. 181(3), 483–492 (2010)CrossRef

Alexandrino, M.M., Briquet, R., Töben, D.: Progress in the theory of singular Riemannian foliations. Differ. Geom. Appl. 31(2), 248–267 (2013)MathSciNetCrossRefMATH

Cecil, T.E.: Isoparametric and Dupin hypersurfaces, SIGMA 4 (2008), Paper 062, 28 pages

Cecil, T.E., Chi, Q.S., Jensen, G.R.: Isoparametric hypersurfaces with four principal curvatures. Ann. Math. 166(1), 1–76 (2007)MathSciNetCrossRefMATH

Cerf, J.: La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie. Inst. Hautes Études Sci. Publ. Math. 39, 5–173 (1970)MathSciNetCrossRefMATH

Chi, Q.S.: The isoparametric story. National Taiwan University, June 25–July 6 (2012). http://www.math.wustl.edu/chi/SummerCourse.pdf

Chi, Q.S.: Isoparametric hypersurfaces with four principal curvatures. III. J. Differ. Geom. 94, 487–504 (2013)MathSciNetMATH

De Sapio, R., Walschap, G.: Diffeomorphism of total spaces and equivalence of bundles. Topology 39, 921–929 (2000)MathSciNetCrossRefMATH

Domínguez-Vázquez, M.: Isoparametric foliations on complex projective spaces. Trans. Am. Math. Soc. 368, 1211–1249 (2016). doi:10.1090/S0002-9947-2014-06415-5

10.

Domínguez-Vázquez, M., Gorodski, C.: Polar foliations on quaternionic projective spaces, preprint (2015). arXiv:1507.02720

11.

Fintushel, R.: Circle actions on simply connected \(4\)-manifolds. Trans. Am. Math. Soc. 230, 147–171 (1977)MathSciNetMATH

12.

Fintushel, R.: Classification of circle actions on \(4\)-manifolds. Trans. Am. Math. Soc. 242, 377–390 (1978)MathSciNetMATH

13.

Freedman, M., Quinn, F.: Topology of \(4\)-manifolds, Princeton Mathematical Series 39. Princeton University Press, Princeton (1990)

14.

Galaz-Garcia, F., Radeschi, M.: Singular Riemannian foliations and applications to positive and nonnegative curvature. J. Topol. 8, 603–620 (2015)MathSciNetCrossRefMATH

15.

Ge, J.Q.: Isoparametric foliations, diffeomorphism groups and exotic smooth structures, preprint (2014). arXiv:1404.6194

16.

Ge, J.Q., Radeschi, M.: Differentiable classification of \(4\)-manifolds with singular Riemannian foliations. Math. Ann. 363, 525–548 (2015)MathSciNetCrossRefMATH

17.

Ge, J.Q., Tang, Z.Z.: Chern conjecture and isoparametric hypersurfaces, in differential geometry—under the influence of S.S. Chern. In: Shen, Y.B., Shen, Z.M., Yau, S.T. Higher Education Press and International Press, Boston (2012)

18.

Ge, J.Q., Tang, Z.Z.: Isoparametric functions and exotic spheres. J. Reine Angew. Math. 683, 161–180 (2013)MathSciNetMATH

19.

Ge, J.Q., Tang, Z.Z., Yan, W.J.: A filtration for isoparametric hypersurfaces in Riemannian manifolds. J. Math. Soc. Jpn. 67, 1179–1212 (2015)MathSciNetCrossRefMATH

20.

Gromoll, D., Walschap, G.: Metric Foliations And Curvature, Progress in Mathematics, 268. Birkhäuser Verlag, Basel (2009)CrossRefMATH

21.

Grove, K., Halperin, S.: Dupin hypersurfaces, group actions and the double mapping cylinder. J. Differ. Geom. 26, 429–459 (1987)MathSciNetMATH

22.

Grove, K., Wilking, B.: A knot characterization and 1-connected nonnegatively curved 4-manifolds with circle symmetry. Geom. Topol. 18, 3091–3110 (2014)MathSciNetCrossRefMATH

23.

Grove, K., Ziller, W.: Curvature and symmetry of Milnor spheres. Ann. Math. 152, 331–367 (2000)MathSciNetCrossRefMATH

24.

Hoelscher, C.A.: Classification of cohomogeneity one manifolds in low dimensions. Pac. J. Math. 246, 129–185 (2010)MathSciNetCrossRefMATH

25.

Joachim, M., Wraith, D.J.: Exotic spheres and curvature. Bull. Am. Math. Soc. (N.S.) 45(4), 595–616 (2008)

26.

Kervaire, M.A., Milnor, J.: Group of homotopy spheres: I. Ann. Math. 77, 504–537 (1963)MathSciNetCrossRefMATH

27.

Levine, J.: Inertia groups of manifolds and diffeomorphisms of spheres. Am. J. Math. 92, 243–258 (1970)MathSciNetCrossRefMATH

28.

Lytchak, A., Wilking, B.: Riemannian foliations of spheres, preprint (2013). arXiv:1309.7884

29.

Milnor, J.: On manifolds homeomorphic to the \(7\)-sphere. Ann. Math. 64(2), 399–405 (1956)

30.

Miyaoka, R.: Isoparametric hypersurfaces with \((g, m)=(6,2)\). Ann. Math. 177, 53–110 (2013)MathSciNetCrossRefMATH

31.

Molino, P.: Riemannian foliations, translated from the French by Grant Cairns. With appendices by G. Cairns, Y. Carrière, E. Ghys, E. Salem, V. Sergiescu. Progress in Math. 73. Birkhäuser Boston Inc, Boston (1988)

32.

Orlik, P., Raymond, F.: Actions of the torus on \(4\)-manifolds I. Trans. Am. Math. Soc. 152, 531–559 (1970)MathSciNetMATH

33.

Orlik, P., Raymond, F.: Actions of the torus on \(4\)-manifolds. II. Topology 13, 89–112 (1974)MathSciNetCrossRefMATH

34.

Palais, R.S.: Extending diffeomorphisms. Proc. Am. Math. Soc. 11, 274–277 (1960)MathSciNetCrossRefMATH

35.

Parker, J.: \(4\)-dimensional G-manifolds with \(3\)-dimensional orbits. Pac. J. Math. 125, 187–204 (1986)MathSciNetCrossRefMATH

36.

Qian, C., Tang, Z.Z.: Isoparametric functions on exotic spheres. Adv. Math. 272, 611–629 (2015)MathSciNetCrossRefMATH

37.

Qian, C., Tang, Z.Z.: Recent progress in isoparametric functions and isoparametric hypersurfaces. Real and complex submanifolds. Springer Proc. Math. Stat. 106, 65–76 (2014)

38.

Radeschi, M.: Clifford algebras and new singular Riemannian foliations in spheres. Geom. Funct. Anal. 24, 1660–1682 (2014)MathSciNetCrossRef

39.

Schultz, R.: On the inertia group of a product of spheres. Trans. Am. Math. Soc. 156, 137–153 (1971)MathSciNetCrossRefMATH

40.

Tang, Z.Z., Xie, Y.Q., Yan, W.J.: Isoparametric foliation and Yau conjecture on the first eigenvalue. II. J. Funct. Anal. 266, 6174–6199 (2014)MathSciNetCrossRefMATH

41.

Tang, Z.Z., Yan, W.J.: Isoparametric foliation and Yau conjecture on the first eigenvalue. J. Differ. Geom. 94, 521–540 (2013)MathSciNetMATH

42.

Tang, Z.Z., Yan, W.J.: Isoparametric foliation and a problem of Besse on generalizations of Einstein condition. Adv. Math. 285, 1970–2000 (2015)MathSciNetCrossRefMATH

43.

Thorbergsson, G.: Isoparametric foliations and their buildings. Ann. Math. 133(2), 429–446 (1991)

44.

Thorbergsson, G.: A survey on isoparametric hypersurfaces and their generalizations. In: Handbook of Differential Geometry, vol. I, pp. 963–995. North-Holland, Amsterdam (2000)

45.

Thorbergsson, G.: Singular Riemannian foliations and isoparametric submanifolds. Milan J. Math. 78, 355–370 (2010)MathSciNetCrossRefMATH

46.

Wang, Q.M.: Isoparametric functions on Riemannian manifolds. I. Math. Ann. 277, 639–646 (1987)MathSciNetCrossRefMATH

Title: Differential Topology Interacts with Isoparametric Foliations
Authors: Chao Qian
Jianquan Ge
Publisher: Springer Japan
Book: Geometry and Topology of Manifolds
Print ISBN: 978-4-431-56019-7

Electronic ISBN: 978-4-431-56021-0

Copyright Year: 2016
DOI: https://doi.org/10.1007/978-4-431-56021-0_8

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Premium Partner