I'm neither an expert in probability theory or in differential geometry. But for a paper I've been reading I'm trying to formalize some concept by myself. This is idea I'm trying to formalize.
Suppose $\mathcal{S} \subset \mathbb{R}^3$ is a surface, and for all $\bf{p} \in \mathcal{S}$ let's define $\mathcal{G}(\bf{p})$ the Gauss map, namely it maps the point $\bf{p}$ to it's normal. Suppose we can partition $\mathcal{S}$ as $$ \mathcal{S} = \bigcup_{j\in J} \mathcal{S_j} $$
suppose the surface $\mathcal{S}$ is finite, and that $J$ is at most countable, you can assume anyway that each patch $\mathcal{S}_j$ is really small. Let's define $\mathcal{G}_j$ the Gauss map previously defined but restricted to $\mathcal{S}_j$. I'm trying to formally define the probability density of $\mathcal{G}_j$ given the distribution of a point $\bf{p}$ in $\mathcal{S}_j$. Assuming $j$ is fixed, the probability of having a certain normal $\textbf{n} \in \mathcal{G}_j(\mathcal{S}_j)$ is given by
$$ p_{\textbf{N}}(\textbf{n}) \left| d \textbf{n} \right| = \sum_{k} p_{\textbf{P}}(\textbf{p}_k) \left|d \textbf{p}_k \right| \Rightarrow p_{\textbf{N}}(\textbf{n}) = \sum_{k} \frac{p_{\textbf{P}}(\textbf{p}_k)}{\frac{\left| d \textbf{n} \right|}{\left|d \textbf{p}_k \right|}} $$
So I need to focus on the computation of
$$ \left| \frac{ d \textbf{n}}{d \textbf{p}_k } \right| = \left| \frac{ d \mathcal{G}(\textbf{p}_k)}{d \textbf{p}_k } \right| $$
Here I'm not sure what I'm doing is going to be correct my gut feeling tells me that
$$ \frac{ d \mathcal{G}(\textbf{p}_k)}{d \textbf{p}_k } = D_{\textbf{p}_k} \mathcal{G}_j $$
However by definition this is equal to the opposite of the Weingarten map, namely
$$ D_{\textbf{p}_k} \mathcal{G}_j = - \mathcal{W}_{\textbf{p}_k,\mathcal{S}_j}, $$
however I have the feeling I'm missing something. The question is if this approach can lead me to some formal definition of probability distributions of normal (as function of the point $\textbf{p}$, or equivalently of the parametrization of the surface $\mathcal{S_j}$. The idea would be given my hypothesis if there's any chance I can express this probability distribution when $\mathcal{p}$ belongs to a very small surface patch (say $d\mathcal{S}$).
Also I'm not entirely sure of my use of differentials here and there...