Let $M, N \subset \mathbb{R}^n$ be two manifolds such that, for every $p \in M \cap N$, $(T_pM)^\bot \cap (T_pN)^\bot = \{0\}$.
How do I determine the tangent space of $M \cap N$? I found some places which stated that the tangent space to the intersection is the intersection of the tangent spaces, but I think that there is a little bit more to it due to the condition on the intersection of the orthogonal spaces.
Any hints?
Note that the condition implies that
$$ \{ 0 \} = (T_pM)^{\perp} \cap (T_pN)^{\perp} = (T_pM + T_pN)^{\perp} $$
and so $T_pM + T_pN = T_p(\mathbb{R}^n) \approx \mathbb{R}^n$. In this case, we say that the subspaces (and if it holds for all $p \in M \cap N$, the manifolds) intersect transversally. Using the implicit function theorem, you can show that this condition guarantees that the intersection $M \cap N$ is indeed a submanifold of dimension $\dim T_pM \cap T_pN$ and with tangent space $T_pM \cap T_pN$. Without this condition, it is not in general true that the intersection $M \cap N$ is even a manifold.