Abstract
This paper continues the investigation of isoparametric triple systems of FKM-type started in [9]. We classify all such triple systems which are congruent to an isoparametric triple system of algebra type. We also consider the question to what extent the Clifford sphere of an FKM-triple is determined by the triple product. As an application the automorphism group of an FKM-triple is described.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M.F. Atiyah, R. Bott, A. Shapiro: Clifford Modules, Topology Vol. 3, Suppl. 1, 3–38.
E. Cartan: Sur des familles remarquables d'hypersurfaces isoparamétriques dans les espaces sphériques, Math. Z.45, 335–367 (1939).
E. Cartan: Sur quelques familles remarquables d'hypersurfaces, C.R. Congrès Math. Liège 30–41, (1939).
E. Cartan: Sur des familles remarquables d'hypersurfaces isoparamétriques des espaces sphériques á 5 et 9 dimensions, Revista Univ. Tucuman, série A,1, 5–22 (1940).
C. Chevalley: The Algebraic Theory of Spinors, Columbia University Press, (1954).
J. Dorfmeister, E. Neher: An algebraic approach to isoparametric hypersurfaces in spheres I, to appear in Tôhoku Math. J.
J. Dorfmeister, E. Neher: An algebraic approach to isoparametric hypersurfaces in spheres II, to appear in Tôhoku Math. J.
J. Dorfmeister, E. Neher: Isoparametric triple systems of algebra type, to appear in Osaka J. Math.
J. Dorfmeister, E. Neher: Isoparametric triple systems of FKM-type I, to appear in Abh. Math. Sem. Hamburg.
J. Dorfmeister, E. Neher: Isoparametric triple systems of FKM-type III, to appear.
D. Ferus: Notes on isoparametric hypersurfaces, Escolo de Geometria Differencial, Universidade Estadual de Campainas (1980).
D. Ferus, H. Karcher, H.F. Münzner: Cliffordalgebren und neue isoparametrische Hyperflächen, Math. Z.177, 479–502 (1981).
A. Hurwitz: Über die Komposition der quadratischen Formen von beliebig vielen Variablen, Mathematische Werke, vol. II, 565–571, Basel: Birkhäuser 1932.
N. Jacobson: Composition algebras and their automorphisms, Rend. Circ. Math. Palermo 7, 55–80 (1958).
N. Jacobson: Structure and representations of Jordan algebras, Amer. Math. Soc. Coll. Publications XXXIX, Providence, Rhode Island 1968.
H.F. Münzner: Isoparametrische Hyperflächen in Sphären I, Math. Ann.251, 57–71 (1980).
H.F. Münzner: Isoparametrische Hyperflächen in Sphären II, Math. Ann.256, 215–232 (1981).
K. Nomizu: Some results in E. Cartan's theory of isoparametric families of hypersurfaces, Bull. Amer. Math. Soc.79, 1184–1188 (1973/4).
K. Nomizu: Elie Cartan's work on isoparametric hypersurfaces, Diff. Geom. Symp. Pure Math.27, II, Standford, 1973.
H. Ozeki, M. Takeuchi: On some types of isoparametric hypersurfaces in spheres I, Tôhoku Math. J.27 515–559 (1975).
H. Ozeki, M. Takeuchi: On some types of isoparametric hypersurfaces in spheres II, Tôhoku Math. J.,28 7–55 (1976).
R. Takagi, T. Takahashi: On the principal curvatures of homogeneous hypersurfaces in a sphere, in: Differential Geometry in honor of K. Yano Tokyo 1972, 469–481.
Author information
Authors and Affiliations
Additional information
Research partially supported by NSF-grant MCS -8104848
Research supported by Deutsche Forschungsgemeinschaft
Rights and permissions
About this article
Cite this article
Dorfmeister, J., Neher, E. Isoparametric triple systems of FKM-type II. Manuscripta Math 43, 13–44 (1983). https://doi.org/10.1007/BF01169095
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01169095