Hypersurface and Foliation Isoparametric Transfer for Generalized $(α,β)$-Metrics

Let $F=αΦ(q,s)$ be a regular generalized $(α,β)$-metric, where $q=\|β\|_α^2$ and $s=β/α$, endowed with the Busemann--Hausdorff or Holmes--Thompson volume. We derive explicit comparison formulas for the induced hypersurface volume and the associated volume mean curvature and use them to study two transfer problems suggested by the Randers case. The first asks whether the same hypersurface is isoparametric for $α$ and $F$; the second asks whether the same regular foliation is isoparametric for both metrics. Exact local and global criteria are obtained. Their Randers specializations recover the corresponding hypersurface transfer and the condition $df(W)=φ(f)$. We also prove an all-normal induced-volume rigidity theorem and construct a complete non-Randers example realizing both transfer mechanisms.