Gauss–Kronecker curvature and equisingularity at infinity of definable families

Assume given a polynomially bounded $o$-minimal structure expanding the real numbers. Let $(T_s)_{s \in \mathbb{R}}$ be a definable 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 $s \to {\lvert K \rvert} (s)$ 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.

Keywords

Gauss–Kronecker curvature, total curvatures, generalized critical values, definable families

2010 Mathematics Subject Classification

Primary 14P10. Secondary 03C64, 57R70.

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The first-named author was partially supported by the ANR project LISA 17-CE400023-01.

The second-named author was supported by CNPq-Brazil grant 150555/2011-3 and by FUNCAP/CAPES/CNPq-Brazil grant 305614/2015-0, and was partially supported by the ANR project LISA 17-CE400023-01.

Received 26 February 2019

Accepted 15 July 2021

Published 24 October 2022