Curve in union of hyperplanes

Question

If a smooth curve $\gamma: [0,T] \to \mathbb R^n$ is contained in the union of hyperplanes $$ \bigcup_{i=1, \dots, N} H_i$$ does it then follow that one can always find time intervals $[t_0, t_1]$ sufficiently small, so that $\gamma(t)$ is contained in one of the hyperplanes, say $H_{i*}$ ?

Answer 1

You can easily construct a $C^\infty$ function $f:(-1,1)\to\mathbb R$ such that $f$ vanishes in each interval of the form $[1/3^n,2/3^n]$ or $[-2/3^n,-1/3^n]$with $n\in\mathbb N$ and nowhere else.

Use it to construct a smooth curve in $(-1,1)\to\mathbb R^3$ with image contained in the union of the $xy$- and $xz$-planes such that no open set containing $0$ is mapped into one of those planes.

Hint: use the function $f$ so that your curves spends the time in the intervals of the form $[2/3^{n+1},1/3^n]$ with $n$ odd in one of the planes and those with $n$ even in the other plane.

Answer 2

If there is a $t$ such that $\sigma(t)$ is in exactly one of the hyperplanes, then clearly a small neighborhood of that $t$ works: this is immediate, by continuity. Indeed, the complement of the union of the other hyperplanes is an open set of $\mathbb R^n$, so its preimage by $\sigma$ is an open set of $\mathbb R$ which contains the point $t$.

If there is no $t$ with that property, then the curve lives in the finite union of lower dimensional affine subspaces which you can get by intersecting the hyperplanes, and you can proceed inductively.