Let $\displaystyle S\in \mathbb{R}^3$ be a connected regular surface. Let $\displaystyle N:S \rightarrow S^2$ be a Gauss map. I am trying to show that $\displaystyle -N$ is the only other Gauss map.

Let $\displaystyle \tilde{N} :S \rightarrow S^2$ be another Gauss map. Is this argument OK?

For $\displaystyle p \in S$,

$\displaystyle \tilde{N}(p) = f(p)N(p)$ where $\displaystyle f(S) = \{-1,1\}$, so

$\displaystyle \tilde{N}$ continuous $\displaystyle \implies f$ continuous $\displaystyle \implies f$ locally constant $\displaystyle \implies f$ constant by connectedness. This means $\displaystyle \tilde{N} = +N$ or $\displaystyle \tilde{N} = -N$.

Thanks for any advice on this!