I am working on a question on closed curves but get stuck on a technical question. Suppose $x(t)=\left(x_1(t), x_2(t), x_3(t)\right):[0,2 \pi] \rightarrow \mathbb{R}^3$ is a smooth closed curve satisfying $x_i(t)=$ $\sum_{n=1}^2\left(c_{i, n} \cos (n t)+s_{i, n} \sin (n t)\right)$ and $\sum_{i=1}^3 x_i(t)^2\equiv 1$ (i.e. $x(t)$ lies on the unit sphere). Does it follow that there exists $n \in\{1,2\}$ such that $c_{i, n}=s_{i, n}=0$ for all $i$? (If $x(t)$ is a plane curve instead it is true.)
One idea is that by considering the Fourier expansion of $1=\sum_{i=1}^3 x_i(t)^2$, we can obtain 9 equations, since the higher Fourier modes of a constant function vanish. So, we have a system of 9 quadratic equations in 6 unknowns. But the system is quite complicated, and I have no clue how to solve it.
After some tedious calculation (done partly by Mathematica), I was able to prove it myself.
By a long but straightforward computation, we have
\begin{align*} \sum_{i=1}^3 x_i(t)^2=& \frac{1}{2} \sum_{i=1}^3 \sum_{n=1}^2\left(c_{i n}^2+s_{i n}^2\right) \\ &+\left(\sum_{i=1}^3 c_{i 1} c_{i 2}+\sum_{i=1}^3 s_{i 1} s_{i 2}\right) \cos (t)+\frac{1}{2}\left(\sum_{i=1}^3 c_{i 1}^2-\sum_{i=1}^3 s_{i 1}^2\right) \cos (2 t) \\ &+\left(\sum_{i=1}^3 c_{i 1} c_{i 2}-\sum_{i=1}^3 s_{i 1} s_{i 2}\right) \cos (3 t)+\frac{1}{2}\left(\sum_{i=1}^3 c_{i 2}^2-\sum_{i=1}^3 s_{i 2}^2\right) \cos (4 t) \\ &+\left(\sum_{i=1}^3\left(-c_{i 2} s_{i 1}+c_{i 1} s_{i 2}\right)\right) \sin (t)+\left(\sum_{i=1}^3 c_{i 1} s_{i 1}\right) \sin (2 t) \\ &+\left(\sum_{i=1}^3\left(c_{i 2} s_{i 1}+c_{i 1} s_{i 2}\right)\right) \sin (3 t)+\left(\sum_{i=1}^3 c_{i_2} s_{i_2}\right) \sin (4 t). \end{align*}
As the LHS of the above is 1, by uniqueness of Fourier series, all the Fourier coefficients except the first one are zero. Let $C_n=\left(c_{1 n}, c_{2 n}, c_{3 n}\right), S_n=\left(s_{1 n}, s_{2 n}, s_{3 n}\right) \in \mathbb{R}^3$ for $n=1,2$. Then from the above, we get \begin{align*} &C_1 \cdot C_2=0, \; S_1 \cdot S_2=0, \; C_2 \cdot S_1=0, \; C_1 \cdot S_2=0, \; C_1 \cdot S_1=0, \; C_2 \cdot S_2=0, \\ &|C_1|^2=|S_1|^2, \; |C_2|^2=|S_2|^2. \end{align*} We claim that either $C_1=S_1=0$ or $C_2=S_2=0$. If not, then from the last two equations above, all $C_1, S_1, C_2, S_2$ are non-zero. But then from the orthogonal relations in the first 6 equations above, we have that $\left\{C_1, C_2, S_1, S_2\right\}$ forms an orthogonal basis for $\mathbb{R}^3$, a contradiction.