Equivalence of Milnor and Milnor-Lê fibrations for real analytic maps

Abstract

In [J. Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61 (Princeton University Press, Princeton, NJ, 1968).] Milnor proved that a real analytic map [Formula: see text], where [Formula: see text], with an isolated critical point at the origin has a fibration on the tube [Formula: see text]. Constructing a vector field such that (1) it is transverse to the spheres, and (2) it is transverse to the tubes, he “inflates” the tube to the sphere, to get a fibration [Formula: see text], but the projection is not necessarily given by [Formula: see text] as in the complex case. In the case [Formula: see text] has isolated critical value, in [J. L. Cisneros-Molina, J. Seade and J. Snoussi, Milnor fibrations and [Formula: see text]-regularity for real analytic singularities, Internat. J. Math. 21(4) (2010) 419–434.] it was proved that if the fibers inside a small tube are transverse to the sphere [Formula: see text], then it has a fibration on the tube. Also in [J. L. Cisneros-Molina, J. Seade and J. Snoussi, Milnor fibrations and [Formula: see text]-regularity for real analytic singularities, Internat. J. Math. 21(4) (2010) 419–434.], the concept of [Formula: see text]-regularity was defined, it turns out that [Formula: see text] is [Formula: see text]-regular if and only if the map [Formula: see text] is a fiber bundle equivalent to the one on the tube. In a more general setting, the corresponding facts are proved in [J. L. Cisneros-Molina, A. Menegon, J. Seade and J. Snoussi, Fibration theorems and [Formula: see text]-regularity for differentiable maps-germs with non-isolated critical value, Preprint (2017).], showing that if a locally surjective map [Formula: see text] has a linear discriminant [Formula: see text] with isolated singularity and a fibration on the tube [Formula: see text], then [Formula: see text] is [Formula: see text]-regular if and only if the map [Formula: see text] (with [Formula: see text] the radial projection of [Formula: see text] on [Formula: see text]) is a fiber bundle equivalent to the one on the tube. In this paper, we generalize this result for an arbitrary linear discriminant by constructing a vector field [Formula: see text] which inflates the tube to the sphere in a controlled way, it satisfies properties analogous to the vector field constructed by Milnor in the complex setting: besides satisfying (1) and (2) above, it also satisfies that [Formula: see text] is constant on the integral curves of [Formula: see text].