Assume that $B$ is a manifold with dimension $2m$ for some $m \in \mathbb{N}$. Let $N$ be sub-manifold with boundary ( denote the boundary by $\partial N$) of $B$ of dimension $m+1$ and let $M$ be a submanifold of $B$ (without boundary) of dimension $m$. Assume that $M \cap int(N) \neq \emptyset$ (where $int(N)$ denotes $N- \partial N$) ,$\partial N \cap M$ is exactly one point and $M$ is not fully contained in $N$.
My question is: is the following claim correct ?
The intersection $M \cap N$ is 1-dimensional sub-manifold near the boundary $\partial N$ (that is there is $V$ a neighborhood of $\partial N$ in $B$ such that $V \cap M \cap N$ is a sub-manifold)
This seems to be correct if we assume that $M$ is a curve (one could convince himself by drawing and see that curves intersect the boundary of $N$ transversely) but what about the general case ?
Edit: Denote by $p$ the unique point of $M \cap \partial N$. What if we assume that the intersection of $T_{p}N$ and $T_{p}M$ is one line ?