A generalization of Cartan's theorem on isoparametric cubics

Abstract

We give a generalization of the well-known result of E. Cartan on isoparametric cubics by showing that a homogeneous cubic polynomial solution of the eiconal equation $|\nabla f{|}^2=9|x{|}^4$ must be rotationally equivalent to either $x_n^3-3x_n(x_1^2+...+x_{n-1}^2)$, or to one of four exceptional Cartan cubic polynomials in dimensions $n=5,8,14,26$.

7 pages, corrected typos; to appear in the Proceedings of the AMS

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