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

Journal of Singularities
volume 13 (2015), 87-106

Received 29 July 2013. Received in revised form 21 February 2014.

DOI: 10.5427/jsing.2015.13e

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Abstract:

Let (X,0), a subset of (R^n,0), be the germ of a closed subanalytic set and consider two subanalytic functions f and g:(X,0) ->(R,0). Under some conditions, we relate the critical points of g on the real Milnor fibre f^{-1}(\delta) \cap B_\epsilon, 0 < |\delta | << \epsilon << 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.

Author(s) information:

Nicolas Dutertre
Aix-Marseille Université, CNRS
Centrale Marseille, I2M, UMR 7373
13453 Marseille, France
email: nicolas.dutertre@univ-amu.fr