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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
DOI: https://doi.org/10.1090/S0002-9939-10-10385-2
Published electronically: March 29, 2010
MathSciNet review: 2644901
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Abstract | References | Similar Articles | Additional Information

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$.


References [Enhancements On Off] (What's this?)

  • 1. É. Cartan, Sur des familles remarquables d'hypersurfaces isoparamétriques dans les espaces sphériques, Math. Z. 45(1939), 335-367. MR 0000169 (1:28f)
  • 2. W. Y. Hsiang, Remarks on closed minimal submanifolds in the standard Riemannian $ m$-sphere. J. Diff. Geom. 1(1967), 257-267. MR 0225244 (37:838)
  • 3. H. B. Lawson, Complete minimal surfaces in $ S\sp{3}$, Ann. of Math. (2) 92(1970), 335-374. MR 0270280 (42:5170)
  • 4. D. B. Shapiro, Compositions of quadratic forms, de Gruyter Expositions in Mathematics, 33. Walter de Gruyter & Co., Berlin, 2000. MR 1786291 (2002f:11046)
  • 5. V. Tkachev, On a classification of minimal cubics in $ \mathbb{R}^n$, in preparation.
  • 6. P. Yiu, Quadratic forms between Euclidean spheres. Manuscripta Math. 83(1994), no. 2, 171-181. MR 1272181 (95b:55013)
  • 7. R. Wood, Polynomial maps from spheres to spheres. Invent. Math. 5(1968), no. 2, 163-168. MR 0227999 (37:3583)

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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: https://doi.org/10.1090/S0002-9939-10-10385-2
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

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