# Gauss-Kronecker Curvature and equisingularity at infinity of definable families

## TL;DR

This paper investigates the behavior of Gauss-Kronecker curvature in definable families of hypersurfaces within o-minimal structures, establishing continuity properties and criteria for equisingularity at infinity.

## Contribution

It introduces a notion of generalized critical value and proves the continuity of curvature functions near these values, providing a new criterion for equisingularity of polynomial level sets.

## Key findings

- Continuity of total Gauss-Kronecker curvature near non-critical values.
- Continuity of total absolute Gauss-Kronecker curvature near non-critical values.
- A necessary condition for equisingularity of polynomial level sets.

## Abstract

Assume given a polynomially bounded o-minimal structure expanding the real numbers. Let $(T_s)_{s\in \mathbb{R}}$ be a globally definable one parameter family of $C^2$-hypersurfaces of $\mathbb{R}^n$. Upon defining the notion of generalized critical value for such a family we show that the functions $s \to |K(s)|$ and $s\to K(s)$, respectively the total absolute Gauss-Kronecker and total Gauss-Kronecker curvature of $T_s$, 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.