Equi-singularity of real families and Lipschitz Killing curvature densities at infinity

Abstract

Fix an o-minimal structure expanding the ordered field of real numbers. Let $(W_y{)}_{y\in {\mathbb{R}}^s}$ be a definable family of closed subsets of ${\mathbb{R}}^n$ whose total space $W={\cup }_y{W}_y\times y$ is a closed connected $C^2$ definable sub-manifold of ${\mathbb{R}}^n\times {\mathbb{R}}^s$. Let $\phi :W\to {\mathbb{R}}^s$ be the restriction of the projection to the second factor. After defining $K(\phi )$, the set of generalized critical values of $\phi$, showing that they are closed and definable of positive codimension in ${\mathbb{R}}^s$, contain the bifurcation values of $\phi$ and are stable under generic plane sections, we prove that all the Lipschitz-Killing curvature densities at infinity $y\mapsto {\kappa }_i^{\infty }(W_y)$ are continuous functions over ${\mathbb{R}}^s\setminus K(\phi )$. When $W$ is a $C^2$ definable hypersurface of ${\mathbb{R}}^n\times {\mathbb{R}}^s$, we further obtain that the symmetric principal curvature densities at infinity $y\mapsto {\sigma }_i^{\infty }(W_y)$ are continuous functions over ${\mathbb{R}}^s\setminus K(\phi )$.