A short proof of the homogeneity of an isoparametric hypersurface with $(g,m)=(6,2)$

A well-known hypersurface classification theorem states that any isoparametric hypersurface in $S^{13}$ with six principal curvatures is homogeneous. This landmark result was first proved in her Annals paper in 2013 by R. Miyaoka with 58 pages, and with errata in Annals in 2016 with 15 pages. The main purpose of this paper is to provide a concise proof of this theorem through an interplay between topological and geometric insights into the Euler class of an oriented vector bundle.