Curvature Integrals on the Real Milnor Fibre

Let f : Rn+1 → R be a polynomial with an isolated critical point at 0 and let ft : Rn+1 → R be a one-parameter deformation of f . We study the differential geometry of the real Milnor fiber Cε t = f −1 t (0) ∩ B n+1 ε . More precisely, we express the following limits : lim ε→0 lim t→0 1 εk ∫ Cε t sn−k(x) dx, where sn−k is the (n− k)-th symmetric function of curvature, in terms of the following averages of topological degrees : ∫ Gn+1 deg0∇(f|H) dH, where Gn+1 is the Grassmann manifold of k-dimensional planes through the origin of Rn+1. When 0 is an algebraically isolated critical point, we study the limits : lim ε→0 lim t→0 1 εk ∫