Isoparametric submanifolds of hyperbolic spaces

Wu, Bingle

doi:10.1090/S0002-9947-1992-1131078-8

Isoparametric submanifolds of hyperbolic spaces
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Trans. Amer. Math. Soc. 331 (1992), 609-626 Request permission

Abstract:

In this paper we prove a decomposition theorem for isoparametric submanifolds of hyperbolic spaces. And as a consequence we obtain all polar actions on hyperbolic spaces. We also prove that any isoparametric submanifold of infinite dimensional hyperbolic space is either totally geodesic, or finite dimensional.

References

Uwe Abresch, Notwendige Bedingungen für isoparametrische Hyperflächen in Sphären mit mehr als drei verschiedenen Hauptkrümmungen, Bonner Mathematische Schriften [Bonn Mathematical Publications], vol. 146, Universität Bonn, Mathematisches Institut, Bonn, 1982 (German). MR 701112
C. T. Benson and L. C. Grove, Finite reflection groups, Bogden & Quigley, Inc., Publishers, Tarrytown-on-Hudson, N.Y., 1971. MR 0383218
Élie Cartan, Familles de surfaces isoparamétriques dans les espaces à courbure constante, Ann. Mat. Pura Appl. 17 (1938), no. 1, 177–191 (French). MR 1553310, DOI 10.1007/BF02410700
Elie Cartan, Sur des familles remarquables d’hypersurfaces isoparamétriques dans les espaces sphériques, Math. Z. 45 (1939), 335–367 (French). MR 169, DOI 10.1007/BF01580289

Sur quelques familles remarquables d’hypersurfaces

Sur des familles d’hypersurfaces isoparametriques des espaces spheriques a

et a

dimension

1

Lawrence Conlon, A class of variationally complete representations, J. Differential Geometry 7 (1972), 149–160. MR 377954
Lawrence Conlon, Variational completeness and $K$-transversal domains, J. Differential Geometry 5 (1971), 135–147. MR 295252
T. E. Cecil and P. J. Ryan, Tight and taut immersions of manifolds, Research Notes in Mathematics, vol. 107, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 781126
Sheila Carter and Alan West, Generalised Cartan polynomials, J. London Math. Soc. (2) 32 (1985), no. 2, 305–316. MR 811789, DOI 10.1112/jlms/s2-32.2.305
Sheila Carter and Alan West, Isoparametric systems and transnormality, Proc. London Math. Soc. (3) 51 (1985), no. 3, 520–542. MR 805720, DOI 10.1112/plms/s3-51.3.520

Isoparametric and totally focal submanifolds

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
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
Carlos Edgard Harle, Isoparametric families of submanifolds, Bol. Soc. Brasil. Mat. 13 (1982), no. 2, 35–48 (Portuguese). MR 735118, DOI 10.1007/BF02584674
Wu-yi Hsiang and H. Blaine Lawson Jr., Minimal submanifolds of low cohomogeneity, J. Differential Geometry 5 (1971), 1–38. MR 298593
Wu-Yi Hsiang, Richard S. Palais, and Chuu-Lian Terng, The topology of isoparametric submanifolds, J. Differential Geom. 27 (1988), no. 3, 423–460. MR 940113
Serge Lang, Introduction to differentiable manifolds, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155257
Hans Friedrich Münzner, Isoparametrische Hyperflächen in Sphären, Math. Ann. 251 (1980), no. 1, 57–71 (German). MR 583825, DOI 10.1007/BF01420281
Katsumi Nomizu, Élie Cartan’s work on isoparametric families of hypersurfaces, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 1, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 191–200. MR 0423260
Barrett O’Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459–469. MR 200865
Barrett O’Neill, Semi-Riemannian geometry, Pure and Applied Mathematics, vol. 103, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. With applications to relativity. MR 719023
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
Richard S. Palais and Chuu-Lian Terng, A general theory of canonical forms, Trans. Amer. Math. Soc. 300 (1987), no. 2, 771–789. MR 876478, DOI 10.1090/S0002-9947-1987-0876478-4
Richard S. Palais and Chuu-Lian Terng, Critical point theory and submanifold geometry, Lecture Notes in Mathematics, vol. 1353, Springer-Verlag, Berlin, 1988. MR 972503, DOI 10.1007/BFb0087442
Chuu-Lian Terng, Isoparametric submanifolds and their Coxeter groups, J. Differential Geom. 21 (1985), no. 1, 79–107. MR 806704
Chuu-Lian Terng, Convexity theorem for isoparametric submanifolds, Invent. Math. 85 (1986), no. 3, 487–492. MR 848682, DOI 10.1007/BF01390326
Chuu-Lian Terng, Submanifolds with flat normal bundle, Math. Ann. 277 (1987), no. 1, 95–111. MR 884648, DOI 10.1007/BF01457280
Chuu-Lian Terng, Proper Fredholm submanifolds of Hilbert space, J. Differential Geom. 29 (1989), no. 1, 9–47. MR 978074
Gudlaugur Thorbergsson, Isoparametric foliations and their buildings, Ann. of Math. (2) 133 (1991), no. 2, 429–446. MR 1097244, DOI 10.2307/2944343