Thanks for reading my question. Assume we are given two compact (possibly with boundary) $(n-1)$-dimensional $C^{\infty}$ manifolds $M_1$ and $M_2$ embedded in $\mathbb{R}^n$, with induced measure $d\sigma_1$ and $d\sigma_2$. For example two $S^{n-1}$ embedded in $\mathbb{R}^n$. Then whether $d\sigma_1\ast d\sigma_2$ is absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}^n$?

I know there is an old result due to Ragozin here:

which deals with *analytic* submanifolds. So I wonder if his results can be extended to smooth manifolds as well. The only place where the analyticness is used is to show that

$$M_1\times M_2\overset{f}{\rightarrow}\mathbb{R}^n$$

defined by

$$f(x,y)=x+y$$

is an analytic map, so its critical *set*, not critical value, is of null measure.

Thank you very much.