Let $\alpha: I \rightarrow \mathbb{R}^{3}$ be a regular smooth curve. Define the curvature $k$ and torsion $\tau$ of $\alpha$ and derive the Frenet formulae. Give the assumption which must hold for torsion to be well-defined, and state the Fundamental Theorem for curves in $\mathbb{R}^{3}$.

Let $\alpha$ be as above and $\tilde{\alpha}: I \rightarrow \mathbb{R}^{3}$ be another regular smooth curve with curvature $\tilde{k}$ and torsion $\tilde{\tau}$. Suppose $\tilde{k}(s)=k(s) \neq 0$ and $\tilde{\tau}(s)=\tau(s)$ for all $s \in I$, and that there exists a non-empty open subinterval $J \subset I$ such that $\left.\tilde{\alpha}\right|_{J}=\left.\alpha\right|_{J}$. Show that $\tilde{\alpha}=\alpha$.

Now let $S \subset \mathbb{R}^{3}$ be an oriented surface and let $\alpha: I \rightarrow S \subset \mathbb{R}^{3}$ be a regular smooth curve contained in $S$. Define normal curvature and geodesic curvature. When is $\alpha$ a geodesic? Give an example of a surface $S$ and a geodesic $\alpha$ whose normal curvature vanishes identically. Must such a surface $S$ contain a piece of a plane? Can such a geodesic be a simple closed curve? Justify your answers.

Show that if $\alpha$ is a geodesic and the Gaussian curvature of $S$ satisfies $K \geqslant 0$, then we have the inequality $k(s) \leqslant 2|H(\alpha(s))|$, where $H$ denotes the mean curvature of $S$ and $k$ the curvature of $\alpha$. Give an example where this inequality is sharp.