I'm currently struggling with the following problem:
Let $M_1, M_2 \subset \mathbb{R}^3$ be two-dimensional submanifolds of $\mathbb{R}^3$ with $M_1 \cap M_2 \neq 0$, s.t. for all $x \in M_1 \cap M_2$: $$T_x(M_1)^\perp \cap T_x(M_2)^\perp = \{0\} $$ Show that $M_1 \cap M_2$ is a one-dimensional submanifold
It seems that such a problem is commonly solved using transversality, but I can't use it yet. I'm given a hint that I should use the following:
Let $n,k \in \mathbb{N}$, with $1 \le k < n, M \subseteq \mathbb{R}^n$. Then (i) $M$ is a $k$-dimensional submanifold of $\mathbb{R}^n$, (ii) for all $a \in M$ there is an open set $\Omega \subseteq \mathbb{R}^n$ and a mapping $G \in C^1(\Omega ; \mathbb{R}^{n-k})$, s.t. $a \in M \cap \Omega = G^{-1}(0)$ and $\text{rank} DG(x) = n-k$ for all $x \in \Omega$.
Any help would be greatly appreciated.