Isoparametric foliations on complex projective spaces

Domínguez-Vázquez, Miguel

doi:10.1090/S0002-9947-2014-06415-5

Isoparametric foliations on complex projective spaces
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by Miguel Domínguez-Vázquez PDF

Trans. Amer. Math. Soc. 368 (2016), 1211-1249 Request permission

Abstract:

Irreducible isoparametric foliations of arbitrary codimension $q$ on complex projective spaces $\mathbb {C} P^n$ are classified, for $(q,n)\neq (1,15)$. Remarkably, there are noncongruent examples that pull back under the Hopf map to congruent foliations on the sphere. Moreover, there exist many inhomogeneous isoparametric foliations, even of higher codimension. In fact, every irreducible isoparametric foliation on $\mathbb {C} P^n$ is homogeneous if and only if $n+1$ is prime.

The main tool developed in this work is a method to study singular Riemannian foliations with closed leaves on complex projective spaces. This method is based on a certain graph that generalizes extended Vogan diagrams of inner symmetric spaces.

References

U. Abresch, Isoparametric hypersurfaces with four or six distinct principal curvatures. Necessary conditions on the multiplicities, Math. Ann. 264 (1983), no. 3, 283–302. MR 714104, DOI 10.1007/BF01459125
Dmitri V. Alekseevsky and Antonio J. Di Scala, The normal holonomy group of Kähler submanifolds, Proc. London Math. Soc. (3) 89 (2004), no. 1, 193–216. MR 2063664, DOI 10.1112/S0024611504014662
Marcos M. Alexandrino, Singular Riemannian foliations with sections, Illinois J. Math. 48 (2004), no. 4, 1163–1182. MR 2113670
Thomas E. Cecil, Quo-Shin Chi, and Gary R. Jensen, Isoparametric hypersurfaces with four principal curvatures, Ann. of Math. (2) 166 (2007), no. 1, 1–76. MR 2342690, DOI 10.4007/annals.2007.166.1
Quo-Shin Chi, Isoparametric hypersurfaces with four principal curvatures, II, Nagoya Math. J. 204 (2011), 1–18. MR 2863363, DOI 10.1215/00277630-1431813
Quo-Shin Chi, Isoparametric hypersurfaces with four principal curvatures, III, J. Differential Geom. 94 (2013), no. 3, 469–504. MR 3080489
Meng-Kiat Chuah and Jing-Song Huang, Double Vogan diagrams and semisimple symmetric spaces, Trans. Amer. Math. Soc. 362 (2010), no. 4, 1721–1750. MR 2574875, DOI 10.1090/S0002-9947-09-04895-8
Ulrich Christ, Homogeneity of equifocal submanifolds, J. Differential Geom. 62 (2002), no. 1, 1–15. MR 1987374
Jiri Dadok, Polar coordinates induced by actions of compact Lie groups, Trans. Amer. Math. Soc. 288 (1985), no. 1, 125–137. MR 773051, DOI 10.1090/S0002-9947-1985-0773051-1
J.-H. Eschenburg and E. Heintze, On the classification of polar representations, Math. Z. 232 (1999), no. 3, 391–398. MR 1719714, DOI 10.1007/PL00004763
J.-H. Eschenburg and E. Heintze, Polar representations and symmetric spaces, J. Reine Angew. Math. 507 (1999), 93–106. MR 1670274, DOI 10.1515/crll.1999.507.93
Dirk Ferus, Hermann Karcher, and Hans Friedrich Münzner, Cliffordalgebren und neue isoparametrische Hyperflächen, Math. Z. 177 (1981), no. 4, 479–502 (German). MR 624227, DOI 10.1007/BF01219082
Jianquan Ge and Zizhou Tang, Isoparametric functions and exotic spheres, J. Reine Angew. Math. 683 (2013), 161–180. MR 3181552, DOI 10.1515/crelle-2012-0005
Jianquan Ge, Zi Zhou Tang, Wenjiao Yan, A filtration for isoparametric hypersurfaces in Riemannian manifolds, to appear in J. Math. Soc. Japan.
Morikuni Goto and Frank D. Grosshans, Semisimple Lie algebras, Lecture Notes in Pure and Applied Mathematics, Vol. 38, Marcel Dekker, Inc., New York-Basel, 1978. MR 0573070
Werner Greub, Stephen Halperin, and Ray Vanstone, Connections, curvature, and cohomology, Pure and Applied Mathematics, Vol. 47-III, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Volume III: Cohomology of principal bundles and homogeneous spaces. MR 0400275
Ernst Heintze, Xiaobo Liu, and Carlos Olmos, Isoparametric submanifolds and a Chevalley-type restriction theorem, Integrable systems, geometry, and topology, AMS/IP Stud. Adv. Math., vol. 36, Amer. Math. Soc., Providence, RI, 2006, pp. 151–190. MR 2222515, DOI 10.1090/amsip/036/04
Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
Stefan Immervoll, On the classification of isoparametric hypersurfaces with four distinct principal curvatures in spheres, Ann. of Math. (2) 168 (2008), no. 3, 1011–1024. MR 2456889, DOI 10.4007/annals.2008.168.1011
Anthony W. Knapp, Lie groups beyond an introduction, 2nd ed., Progress in Mathematics, vol. 140, Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1920389
H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992
Ottmar Loos, Symmetric spaces. II: Compact spaces and classification, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0239006
Alexander Lytchak, Polar foliations of symmetric spaces, Geom. Funct. Anal. 24 (2014), no. 4, 1298–1315. MR 3248486, DOI 10.1007/s00039-014-0279-2
Reiko Miyaoka, Isoparametric hypersurfaces with $(g,m)=(6,2)$, Ann. of Math. (2) 177 (2013), no. 1, 53–110. MR 2999038, DOI 10.4007/annals.2013.177.1.2
Hans Friedrich Münzner, Isoparametrische Hyperflächen in Sphären, Math. Ann. 251 (1980), no. 1, 57–71 (German). MR 583825, DOI 10.1007/BF01420281
Hideki Ozeki and Masaru Takeuchi, On some types of isoparametric hypersurfaces in spheres. I, Tohoku Math. J. (2) 27 (1975), no. 4, 515–559. MR 454888, DOI 10.2748/tmj/1178240941
Fabio Podestà and Gudlaugur Thorbergsson, Polar actions on rank-one symmetric spaces, J. Differential Geom. 53 (1999), no. 1, 131–175. MR 1776093
Hans Samelson, Notes on Lie algebras, 2nd ed., Universitext, Springer-Verlag, New York, 1990. MR 1056083, DOI 10.1007/978-1-4613-9014-5
Barry Simon, Representations of finite and compact groups, Graduate Studies in Mathematics, vol. 10, American Mathematical Society, Providence, RI, 1996. MR 1363490, DOI 10.1038/383266a0
Stephan Stolz, Multiplicities of Dupin hypersurfaces, Invent. Math. 138 (1999), no. 2, 253–279. MR 1720184, DOI 10.1007/s002220050378
Ryoichi Takagi, A class of hypersurfaces with constant principal curvatures in a sphere, J. Differential Geometry 11 (1976), no. 2, 225–233. MR 425848
Chuu-Lian Terng, Isoparametric submanifolds and their Coxeter groups, J. Differential Geom. 21 (1985), no. 1, 79–107. MR 806704
Chuu-Lian Terng, Submanifolds with flat normal bundle, Math. Ann. 277 (1987), no. 1, 95–111. MR 884648, DOI 10.1007/BF01457280
C.-L. Terng and G. Thorbergsson, Submanifold geometry in symmetric spaces, J. Differential Geom. 42 (1995), no. 3, 665–718. MR 1367405
Gudlaugur Thorbergsson, Isoparametric foliations and their buildings, Ann. of Math. (2) 133 (1991), no. 2, 429–446. MR 1097244, DOI 10.2307/2944343
Gudlaugur Thorbergsson, Singular Riemannian foliations and isoparametric submanifolds, Milan J. Math. 78 (2010), no. 1, 355–370. MR 2684784, DOI 10.1007/s00032-010-0112-9
Qi Ming Wang, Isoparametric hypersurfaces in complex projective spaces, Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Vol. 1, 2, 3 (Beijing, 1980) Sci. Press Beijing, Beijing, 1982, pp. 1509–1523. MR 714387
Bingle Wu, Isoparametric submanifolds of hyperbolic spaces, Trans. Amer. Math. Soc. 331 (1992), no. 2, 609–626. MR 1131078, DOI 10.1090/S0002-9947-1992-1131078-8
Liang Xiao, Classification of irreducible isoparametric submanifolds in $\mathbf C\textrm {P}^n$, Sci. China Ser. A 43 (2000), no. 1, 28–34. MR 1766241, DOI 10.1007/BF02903845
Liang Xiao, Principal curvatures of isoparametric hypersurfaces in $\mathbf C\textrm {P}^n$, Trans. Amer. Math. Soc. 352 (2000), no. 10, 4487–4499. MR 1779484, DOI 10.1090/S0002-9947-00-02578-2
Shing-Tung Yau, Open problems in geometry, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990) Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 1–28. MR 1216573