Cohomogeneity one actions on noncompact symmetric spaces of rank one
HTML articles powered by AMS MathViewer
- by Jürgen Berndt and Hiroshi Tamaru PDF
- Trans. Amer. Math. Soc. 359 (2007), 3425-3438 Request permission
Abstract:
We classify, up to orbit equivalence, all cohomogeneity one actions on the hyperbolic planes over the complex, quaternionic and Cayley numbers, and on the complex hyperbolic spaces $\mathbb C H^n$, $n \geq 3$. For the quaternionic hyperbolic spaces $\mathbb H H^n$, $n \geq 3$, we reduce the classification problem to a problem in quaternionic linear algebra and obtain partial results. For real hyperbolic spaces, this classification problem was essentially solved by Élie Cartan.References
- Dmitri V. Alekseevsky and Antonio J. Di Scala, Minimal homogeneous submanifolds of symmetric spaces, Lie groups and symmetric spaces, Amer. Math. Soc. Transl. Ser. 2, vol. 210, Amer. Math. Soc., Providence, RI, 2003, pp. 11–25. MR 2018350, DOI 10.1090/trans2/210/02
- Jürgen Berndt, Homogeneous hypersurfaces in hyperbolic spaces, Math. Z. 229 (1998), no. 4, 589–600. MR 1664778, DOI 10.1007/PL00004673
- Jürgen Berndt and Martina Brück, Cohomogeneity one actions on hyperbolic spaces, J. Reine Angew. Math. 541 (2001), 209–235. MR 1876290, DOI 10.1515/crll.2001.093
- Jürgen Berndt and Hiroshi Tamaru, Homogeneous codimension one foliations on noncompact symmetric spaces, J. Differential Geom. 63 (2003), no. 1, 1–40. MR 2015258
- Jürgen Berndt and Hiroshi Tamaru, Cohomogeneity one actions on noncompact symmetric spaces with a totally geodesic singular orbit, Tohoku Math. J. (2) 56 (2004), no. 2, 163–177. MR 2053317
- Jürgen Berndt, Franco Tricerri, and Lieven Vanhecke, Generalized Heisenberg groups and Damek-Ricci harmonic spaces, Lecture Notes in Mathematics, vol. 1598, Springer-Verlag, Berlin, 1995. MR 1340192, DOI 10.1007/BFb0076902
- Armand Borel, Le plan projectif des octaves et les sphères comme espaces homogènes, C. R. Acad. Sci. Paris 230 (1950), 1378–1380 (French). MR 34768
- Robert L. Bryant, Submanifolds and special structures on the octonians, J. Differential Geometry 17 (1982), no. 2, 185–232. MR 664494
- M. Buchner, K. Fritzsche, and Takashi Sakai, Geometry and cohomology of certain domains in the complex projective space, J. Reine Angew. Math. 323 (1981), 1–52. MR 611441, DOI 10.1515/crll.1981.323.1
- 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
- Antonio J. Di Scala and Carlos Olmos, The geometry of homogeneous submanifolds of hyperbolic space, Math. Z. 237 (2001), no. 1, 199–209. MR 1836778, DOI 10.1007/PL00004860
- Patrick B. Eberlein, Geometry of nonpositively curved manifolds, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996. MR 1441541
- Reese Harvey and H. Blaine Lawson Jr., Calibrated geometries, Acta Math. 148 (1982), 47–157. MR 666108, DOI 10.1007/BF02392726
- Wu-yi Hsiang and H. Blaine Lawson Jr., Minimal submanifolds of low cohomogeneity, J. Differential Geometry 5 (1971), 1–38. MR 298593
- Koichi Iwata, Classification of compact transformation groups on cohomology quaternion projective spaces with codimension one orbits, Osaka Math. J. 15 (1978), no. 3, 475–508. MR 510490
- Koichi Iwata, Compact transformation groups on rational cohomology Cayley projective planes, Tohoku Math. J. (2) 33 (1981), no. 4, 429–442. MR 643227, DOI 10.2748/tmj/1178229347
- Andreas Kollross, A classification of hyperpolar and cohomogeneity one actions, Trans. Amer. Math. Soc. 354 (2002), no. 2, 571–612. MR 1862559, DOI 10.1090/S0002-9947-01-02803-3
- T. Levi-Civita, Famiglie di superficie isoparametriche nell’ordinario spazio euclideo. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (6) 26 (1937), 355–362.
- A. L. Onishchik, Inclusion relations among transitive compact transformation groups. Amer. Math. Soc. Transl. (2) 50 (1966), 5–58.
- B. Segre, Famiglie di ipersuperficie isoparametriche negli spazi euclidei ad un qualunque numero di dimensioni. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (6) 27 (1938), 203–207.
- Ryoichi Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka Math. J. 10 (1973), 495–506. MR 336660
Additional Information
- Jürgen Berndt
- Affiliation: Department of Mathematics, University College, Cork, Ireland
- Email: j.berndt@ucc.ie
- Hiroshi Tamaru
- Affiliation: Department of Mathematics, Hiroshima University, 1-3-1 Kagamiyama, Higashi- Hiroshima, 739-8526, Japan
- MR Author ID: 645435
- Email: tamaru@math.sci.hiroshima-u.ac.jp
- Received by editor(s): July 12, 2005
- Published electronically: January 26, 2007
- Additional Notes: The second author was partially supported by Grant-in-Aid for Young Scientists (B) 14740049 and 17740039, The Ministry of Education, Culture, Sports, Science and Technology, Japan
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 359 (2007), 3425-3438
- MSC (2000): Primary 53C35; Secondary 57S20
- DOI: https://doi.org/10.1090/S0002-9947-07-04305-X
- MathSciNet review: 2299462