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.

中文翻译：

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可定义族无穷处的高斯-克罗内克曲率和等奇异性

假设给定一个扩展实数的多项式有界$o$-最小结构。$Let (T_s)_{s \in \mathbb{R}}$ 是 $\mathbb{R}^n$ 的 $C^2$-超曲面的可定义族。在定义这样一个族的广义临界值的概念后，我们证明了函数 $s \to {\lvert K \rvert} (s)$ 和 $s \to K(s)$，分别是总绝对高斯– $T_s$ 的 Kronecker 和总 Gauss-Kronecker 曲率在任何非广义临界值的任何邻域中都是连续的。特别是，这为实数多项式的能级族提供了必要的等奇异性标准。