I'm learning Differential Geometry recently with do Carmo's book. In the book, Gauss map is define as a differentiable map from an orientable surface $\mathcal{S}$ to $S^2$ in such a way that for every point on a $\mathcal{S}$ it maps to the unit normal vector at the point.

Here, the differentiable map of unit normal vectors is of course globally defined only if $\mathcal{S}$ is orientable. However, every surface is locally orientable, so why not defining Gauss map locally?

As I learn what curvature of a surface is, I came to think that Gauss made Gauss map in order to define curvature. However, curvature is always defined locally. So there has to be something more that requires globally defined Gauss map...

I finally arrived to this idea: "curvatures such as principal curvatures or mean curvature have sign dependent on the direction of normal. So, in order to define those curvatures continuously on the whole surface, we need an orientable surface. For example, we cannot define principal curvatures or mean curvature for Möbius strip that vary continuously on the whole surface."

I wonder if my idea makes sense?


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