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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A generalization of Cartan's theorem on isoparametric cubics


Author: Vladimir G. Tkachev
Journal: Proc. Amer. Math. Soc. 138 (2010), 2889-2895
MSC (2010): Primary 53C42, 35F20, 17A35; Secondary 15A63, 17A75
Published electronically: March 29, 2010
MathSciNet review: 2644901
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Abstract: We generalize the well-known result of É. Cartan on isoparametric cubics by showing that a homogeneous cubic polynomial solution of the eiconal equation $ \vert\nabla f\vert^2=9\vert x\vert^4$ must be rotationally equivalent to either $ x_n^3-3x_n(x_1^2+\ldots+x_{n-1}^2)$ or to one of four exceptional Cartan cubic polynomials in dimensions $ n=5,8,14,26$.


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Additional Information

Vladimir G. Tkachev
Affiliation: Department of Mathematics, Royal Institute of Technology, SE-10044 Stockholm, Sweden
Email: tkatchev@kth.se

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10385-2
PII: S 0002-9939(10)10385-2
Keywords: Cartan's theorem, division algebras, composition formulas, quadratic maps, eiconal equation, minimal cubic cones
Received by editor(s): August 20, 2009
Published electronically: March 29, 2010
Communicated by: Chuu-Lian Terng
Article copyright: © Copyright 2010 American Mathematical Society