Asian Journal of Mathematics

Volume 25 (2021)

Number 6

Gauss–Kronecker curvature and equisingularity at infinity of definable families

Pages: 815 – 840

DOI: https://dx.doi.org/10.4310/AJM.2021.v25.n6.a2

Authors

Nicolas Dutertre (Université Angers, CNRS, LAREMA, SFR MATHSTIC, Angers, France)

Vincent Grandjean (Departamento de Matemática, Universidade Federal do Ceará (UFC), Campus do Pici, Fortaleza-Ce, Brasil)

Abstract

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.

The full text of this article is unavailable through your IP address: 44.210.77.73

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