Milnor fibrations on rho-tubes and rho-spheres for real analytic map germs

Abstract

For a real analytic map germ $G:(\mathbb{R}^m,0)\to(\mathbb{R}^p,0)$, we study Milnor fibrations obtained by replacing the squared Euclidean distance with an analytic control function $ρ$ defining the origin. We formulate the corresponding Milnor set and condition (b), derive criteria for tube and sphere fibrations, and examine their dependence on $ρ$. The family $G_ρ=(xz,\,yzρ)$ provides explicit changes of regularity under elliptic controls. In particular, the germ $G(x,y,z)=(xz,\,yz(10x^2+y^2+3z^2))$ admits neither the induced tube nor sphere fibration for the Euclidean control, while both fibrations exist for the adapted function $ρ=10x^2+y^2+3z^2$. Thus the control function can be essential to the local fibration structure.