Intersection of non transverse submanifolds

Question

Hi! Probably this is an easy question, but i can't see the answer.

Let $X$ be a a smooth real manifold with $\dim(X)=d$ and $M,N\subset X$ two smooth submanifolds with $\dim(M)=m$ and $\dim(N)=n$. The submanifolds $M,N$ intersect but not transversely.

What can i say about connected components of $M\cap N$? More precisely, is it possible to find three manifolds $X,M,N$ as above such that a connected component of $M\cap N$ is not a manifold? Or a connected component that is not smooth? (In all the examples i thought, connected components of $M\cap N$ were smooth)

Thank you in advance.

Answer 1

Let $M$ be any manifold, and let $Z$ be a closed subset of $M$. Suppose there exists a smooth function $f:M\to\mathbb{R}$ with $f^{-1}\{0\}=Z$. We can then take $X=M\times\mathbb{R}$ and identify $M$ with $M\times\{0\}$ and put $N=\{(m,f(m)):m\in M\}$. Then $M$ and $N$ are embedded submanifolds of $X$ with $M\cap N=Z$.

Moreover, I think it is true that such a function $f$ exists for every closed subset $Z$, no matter how wild or fractal. I don't remember the argument in detail, but if I recall correctly it is not too hard. One issue is to patch together things done locally using a partition of unity, and another is to express $f$ as a countable sum of nonnegative smooth functions $f_n$ which need to be rescaled aggressively to force the higher derivatives of the sum to converge.

Answer 2

You could try the real quadric surface $x^2+y^2-z^2=1$ and the hyperplane $x=1$. They intersect along a union of two (intersecting) lines ($z=\pm y$, $x=1$).