Showing papers by "Maria Aparecida Soares Ruas published in 2002"

On Real Singularities with a Milnor Fibration

Abstract: In this article we study the singularities defined by real analytic maps $$ \left( {\mathbb{R}^m ,0} \right) \to \left( {\mathbb{R}^2 ,0} \right) $$ with an isolated critical point at the origin, having a Milnor fibration. It is known [14] that if such a map has rank 2 on a punctured neighbourhood of the origin, then one has a fibre bundle φ : S m−1 − → S 1, where K is the link. In this case we say that f satisfies the Milnor condition at 0 ∈ ℝ m . However, the map φ may not be the obvious map \( \frac{f} {{\parallel f\parallel }} \) as in the complex case [14, 9]. If f satisfies the Milnor condition at 0 ∈ ℝ m and for every sufficiently small sphere around the origin the map \( \frac{f} {{\parallel f\parallel }} \) defines a fibre bundle, then we say that f satisfies the strong Milnor condition at 0 ∈ ℝ m . In this article we first use well known results of various authors to translate “the Milnor condition” into a problem of finite determinacy of map germs, and we study the stability of these singularities under perturbations by higher order terms. We then complete the classification, started in [20, 21] of certain families of singularities that satisfy the (strong) Milnor condition. The simplest of these are the singularities in ℝ2 n ≅ ℂ n of the form \(\{ \sum _{i = 1}^nz_i^{{a_i}}z_i^{ - {b_i}} = 0, {a_i} > {b_i} \geqslant 1\}\) We prove that these are topologically equivalent (but not analytically equivalent!) to Brieskorn-Pham singularities.