Let $\gamma(t):[a, b] \rightarrow \mathbb{R}^{3}$ denote a regular curve.

(a) Show that there exists a parametrization of $\gamma$ by arc length.

(b) Under the assumption that the curvature is non-zero, define the torsion of $\gamma$. Give an example of two curves $\gamma_{1}$ and $\gamma_{2}$ in $\mathbb{R}^{3}$ whose curvature (as a function of arc length $s$ ) coincides and is non-vanishing, but for which the curves are not related by a rigid motion, i.e. such that $\gamma_{1}(s)$ is not identically $\rho_{(R, T)}\left(\gamma_{2}(s)\right)$ where $R \in S O(3), T \in \mathbb{R}^{3}$ and

$$\rho_{(R, T)}(v):=T+R v$$

(c) Give an example of a simple closed curve $\gamma$, other than a circle, which is preserved by a non-trivial rigid motion, i.e. which satisfies

$$\rho_{(R, T)}(v) \in \gamma([a, b]) \text { for all } v \in \gamma([a, b])$$

for some choice of $R \in S O(3), T \in \mathbb{R}^{3}$ with $(R, T) \neq(\mathrm{Id}, 0)$. Justify your answer.

(d) Now show that a simple closed curve $\gamma$ which is preserved by a nontrivial smooth 1-parameter family of rigid motions is necessarily a circle, i.e. show the following:

Let $(R, T):(-\epsilon, \epsilon) \rightarrow S O(3) \times \mathbb{R}^{3}$ be a regular curve. If for all $\tilde{t} \in(-\epsilon, \epsilon)$,

$$\rho_{(R(\tilde{t}), T(\tilde{t}))}(v) \in \gamma([a, b]) \text { for all } v \in \gamma([a, b]) \text {, }$$

then $\gamma([a, b])$ is a circle. [You may use the fact that the set of fixed points of a non-trivial rigid motion is either $\emptyset$ or a line $L \subset \mathbb{R}^{3}$.]