Let $\alpha:[0, L] \rightarrow \mathbf{R}^{3}$ be a regular curve parametrized by arc length having nowherevanishing curvature. State the Frenet relations between the tangent, normal and binormal vectors at a point, and their derivatives.

Let $S \subset \mathbf{R}^{3}$ be a smooth oriented surface. Define the Gauss map $N: S \rightarrow S^{2}$, and show that its derivative at $P \in S, d N_{P}: T_{P} S \rightarrow T_{P} S$, is self-adjoint. Define the Gaussian curvature of $S$ at $P$.

Now suppose that $\alpha:[0, L] \rightarrow \mathbf{R}^{3}$ has image in $S$ and that its normal curvature is zero for all $s \in[0, L]$. Show that the Gaussian curvature of $S$ at a point $P=\alpha(s)$ of the curve is $K(P)=-\tau(s)^{2}$, where $\tau(s)$ denotes the torsion of the curve.

If $S \subset \mathbf{R}^{3}$ is a standard embedded torus, show that there is a curve on $S$ for which the normal curvature vanishes and the Gaussian curvature of $S$ is zero at all points of the curve.