Brouwer degrees of the gradient maps of isoparametric functions

摘要

According to the geometric definition of the Brouwer degree, Peng and Tang obtained the Brouwer degree of the gradient maps of isoparametric function by means of moving frames and the geometry of isoparametric hypersurfaces, and they gave some new examples of harmonic maps between spheres. In this note we firstly study the relation between the tangent map of Φ and the Hessian of f, decompose the isoparametric polynomial by means of the method in ref., give an explicit form of some part in f, and then calculate the degree of every Φ.