Stratified critical points on the real Milnor fibre and integral-geometric formulas

Abstract

Let $(X,0) \subset (\mathbb{R}^n,0)$ be the germ of a closed subanalytic set and let $f$ and $g : (X,0) \rightarrow (\mathbb{R},0)$ be two subanalytic functions. Under some conditions, we relate the critical points of $g$ on the real Milnor fibre $X \cap f^{-1}(\delta) \cap B_\epsilon$, $0 < \vert \delta \vert \ll \epsilon \ll 1$, to the topology of this fibre and other related subanalytic sets. As an application, when $g$ is a generic linear function, we obtain an ''asymptotic Gauss-Bonnet formula for the real Milnor fibre of $f$. From this Gauss-Bonnet formula, we deduce ''infinitesimal linear kinematic formulas.