I need the following theorem: Let $A$ and $B$ be two $n-1$ dimensional bounded smooth manifolds in $R^{n}$. Let $C$ be the set of all points in $A \cap B$ where the normals to $A$ and $B$ are unequal. Then $C$ has measure zero, in the $n-1$ dimensional Hausdorff measure.


I actually only need the cases $n \leq 4$, and where the normals are actually orthogonal.



Measure of the intersection of two manifolds
This is Thom-Boardman transversality, and is an elementary exercise, since the 
intersection is a submanifold of dimension $n-2$.


A pretty complete reference: <https://en.wikipedia.org/wiki/Transversality_(mathematics)>


