In this note we consider a special case of the famous Coarea Formula whose initial proof (for functions from any Riemannian manifold of dimension 2 into $${\mathbb {R}}$$R) is due to Kronrod (Uspechi Matem Nauk 5(1):24–134, 1950) and whose general proof (for Lipschitz maps between two Riemannian manifolds of dimensions n and p) is due to Federer (Am Math Soc 93:418–491, 1959). See also Maly et al. (Trans Am Math Soc 355(2):477–492, 2002), Fleming and Rishel (Arch Math 11(1):218–222, 1960) and references therein for further generalizations to Sobolev mappings and BV functions respectively. We propose two counterexamples which prove that the coarea formula that we can find in many references (for example Bérard (Spectral geometry: direct and inverse problems, Springer, 1987), Berger et al. (Le Spectre d’une Variété Riemannienne, Springer, 1971) and Gallot (Astérisque 163(164):31–91, 1988), is not valid when applied to $$C^\infty $$C∞ functions. The gap appears only for the non generic set of non Morse functions.

中文翻译：

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关于coarea公式的简要说明

在这篇笔记中，我们考虑著名的 Coarea 公式的一个特例，它的初始证明（对于从任何 2 维黎曼流形到 $${\mathbb {R}}$$R 的函数）是由于 Kronrod (Uspechi Matem Nauk 5( 1):24–134, 1950) 并且其一般证明（对于两个 n 和 p 维的黎曼流形之间的 Lipschitz 映射）归功于 Federer（Am Math Soc 93:418–491, 1959）。另见 Maly 等人。(Trans Am Math Soc 355(2):477–492, 2002), Fleming and Rishel (Arch Math 11(1):218–222, 1960) 以及其中的参考资料，分别进一步概括了 Sobolev 映射和 BV 函数。我们提出了两个反例来证明我们可以在许多参考文献中找到的 coarea 公式（例如 Bérard（光谱几何：直接和逆问题，Springer，1987），Berger 等人（Le Specter d'une Variété Riemannienne，Springer，1971) 和 Gallot (Astérisque 163(164):31–91, 1988)，当应用于 $$C^\infty $$C∞ 函数时无效。差距只出现在非莫尔斯函数的非通用集上。