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 $\rho$ defining the origin. We formulate the corresponding Milnor set and condition (b), derive criteria for tube and sphere fibrations, and examine their dependence on $\rho$. The family $G_{\rho }=(xz,yz\rho )$ 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 $\rho =10x^2+y^2+3z^2$. Thus the control function can be essential to the local fibration structure.

17 pages,