Consider two smooth vector fields $X$, $Y$ defined on a manifold $M$.
I am interesting in the set $L=\{q\in M;\; \langle X(q),Y(q)\rangle=0\}$.
I am wondering what is the geometry of the set $L$.
Intuitively, I see the following cases:
- $L=\emptyset$
- $L$ is a singleton
- $L$ is a union of smooth curves
- $L=M$
but I don't know which case appears generically
In the case $M=\mathbb R^n$ (with $n\ge 2$), $L$ can be literally any closed subset. To see this, let $K\subset\mathbb R^n$ be an arbitrary closed subset, and let $f\colon \mathbb R^n\to \mathbb R$ be a smooth function whose zero set is exactly $K$. (Such a function exists by Theorem 2.29 in my Introduction to Smooth Manifolds, 2nd ed.) Then define $X$ and $Y$ by \begin{align*} X & = \frac{\partial}{\partial x^1},\\ Y & = \frac{\partial}{\partial x^2} + f(x) \frac{\partial}{\partial x^1}. \end{align*} Then $L=K$.
On an arbitrary smooth manifold, there may be some topological restrictions. For example, to obtain vector fields for which $L=\emptyset$, there must be two independent global vector fields, which is not always possible. But $L$ can still be pretty awful -- for example, if $K$ is any closed subset contained in the domain of a coordinate chart $U$, you can always find vector fields for which $L\cap U = K$.