A brief note on the coarea formula

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 \({\mathbb {R}}\)) 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 \(C^\infty \) functions. The gap appears only for the non generic set of non Morse functions.