Gauss map on trace of curve Suppose we have a curve $\gamma : I \to S$ where $I \subseteq \mathbb{R}$ is an interval and $S$ is a surface in $\mathbb{R}^3$ (not necessarily orientable). We know that the Gauss map $N$ can be locally defined at all points of $S$, but there is not necessarily a way to define it continuously on the whole surface. Can we define a map $N_I : I \to S^2$ such that $N_I(t)$ is normal to the surface $S$ at $\gamma(t)$?
Notice that $N_I$ is not necessarily $N|_{\gamma(I)}$, since for two points $t_0, t_1$ with $\gamma(t_0) = \gamma(t_1)$ it is possible that $N_I(t_0) \neq N_I(t_1)$ (indeed, this must happen necessarily if the manifold is non orientable). This question came to me as a variation of the problem 2.6.6 of Do Carmo Differential Geometry of Curves and Surfaces.

Yes, this is possible. Intuitively, at every point $s \in I$ there are two choices of unit normal to $S$ at $\gamma(s)$. Since there are no loops in $I$, it is possible to make a consistent global choice.
Using the language of vector bundles, the pullback of the normal bundle $TS^\perp$ to the interval I, $\gamma^*(TS^\perp) := \{(s,\nu) : s\in I, \, \nu \in T_{\gamma(s)}S^\perp\}$, is isomorphic to the trivial bundle, $I \times \mathbb{R}$. Therefore we may choose a global section $N: I \rightarrow \mathbb{R}$ corresponding to a consistent choice of normal along the curve.
