Isoparametric foliations of $S^n\times S^n$ of codimension $1$

Abstract

In this article, we give several families of new examples of isoparametric foliations of $S^n\times S^n$; establish some basic properties of isoparametric foliations of $S^n\times S^n$ of codimension $1$, especially any of them can be realized as level sets of the restriction of a bi-homogeneous polynomial of bi-degree $(1,1),(2,2),(3,3),(4,4)$ or $(6,6)$ to $S^n\times S^n$; and classify bi-homogeneous isoparametric polynomials of bi-degree $(1,1)$ and $(2,2)$ and isoparametric foliations on $S^n\times S^n$ with $n$ even and not equal to $4$.