i'm studying differential geometry for the first time and I just started reading about the gauss map and I've gotta say this stuff is pretty cool. The claim made in the title of this post is from page 142 of Do Carmo's Differential geometry book, where it is stated as "proposition 1". I have never taken a proper geometry class before and the text is starting to get to the point where the density of the calculus/linear algebra is making the geometric significance of the statements rather opaque. That being said, I feel that there is an 'aha' moment just around the corner with this one, and people here are usually pretty good at breaking things down for me so I thought i'd ask y'll this rather soft question.
Anyway, just to refresh your memory, I'm looking to understand the geometric significance of the differential of the Gauss map, $dN_p: T_p(S) \rightarrow T_p(S)$, being a self adjoint linear map. (in particular, what is the geometric significance of this linear map being 'self adjoint'.)
Where $N: S \rightarrow S^2$ is a map from the set of normal unit vectors of a regualer surface $S$ to the unit sphere $S^2$ and $T_p(S)$ is that tangent plane of an arbitrary $p \in S$.
Also, note that $T_p(S)$ and $T_{N(p)}(S^2)$ are the same as vector spaces, which is why we can view $dN_p$ as a linear map from $T_p(S) \rightarrow T_p(S)$