Gauss-Kronecker Curvature and equisingularity at infinity of definable families

Abstract : Assume given a polynomially bounded o-minimal structure expanding the real numbers. Let (Ts)s∈R be a globally definable family of C2 hypersurfaces of Rn. Upon defining the notion of generalized critical value for such a family, we show that the functions s → |K(s)| and s → K(s), respectively the total absolute Gauss-Kronecker and total Gauss-Kronecker curvature of Ts, are continuous in any neighbourhood of any value which is not generalized critical. In particular this provides a necessary criterion of equisingularity for the family of the levels of a real polynomial