Take $M$ to be a manifold and $N_{1},N_{2} \subset M$ to be two submanifolds such that $dimN_{1}+dimN_{2}=dimM$. Then $N_{1}$ and $N_{2}$ are transverse if and only if their intersection is infinite discrete (i.e. contains either finitely or countably infinite many elements) and $TN_{1} \bigoplus TN_{2} \cong TM $ at any point $p \in N_{1} \cap N_{2}$.
Recall that any two $N_{1},N_{2}$ are transverse submanifolds of $M$ if $T_{x}N_{1} + T_{x}N_{2}=T_{x}M$ for all $x \in N_{1} \cap N_{2}$.
Suppose that $M=\mathbb{R}^{3}$ and take $N_{1}$ and $N_{2}$ to be such that $dimN_{1}+dimN_{2}=3$ but they are not transverse.
More precisely I would like to see an example where $TN_{1} \bigoplus TN_{2} \cong TM$ fails.
Is there a concrete example of such situation, maybe $S^{2}$ and the circle $S^{1}$ touching at one unique point?
Thanks in advance!
One way to do this is to start in $\mathbb{R}^{2}$ and then move one dimension up.
In $\mathbb{R}^{2}$, consider the line $\ell = \{(x,y) \, \mid \, y = 0\}$. Here are two curves $\gamma$ that intersect it at a single point, but not transversely: $\gamma_{1} = \{(x,x^{2}) \, \mid \, x \in \mathbb{R}\}$ and $\gamma_{2} = \{(x,x^{3}) \, \mid \, x \in \mathbb{R}\}$.
Now move to $\mathbb{R}^{3}$ in the following way. Let $P$ be the plane $P = \{(x,y,z) \, \mid \, y = 0\} = \pi^{-1}(\ell)$. (Here $\pi : \mathbb{R}^{3} \to \mathbb{R}^{2}$ is the projection $\pi(x,y,z) = (x,y)$.) Next, let $\tilde{\gamma}$ be the curve $\tilde{\gamma} = \{(x,y,0) \, \mid (x,y) \in \gamma\}$.
We have $\text{dim} \, P + \text{dim} \, \tilde{\gamma} = 3$. However, they do not intersect transversely.