Submission date: 20 March 2023

Abstract:

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

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Full text arXiv 2303.11474: pdf, ps.