Suppose there are two orientable surfaces $M^1$ and $M^2$ with their respective gaussian maps $N^1$, $N^2$. Such that $p\in M^1\cap M^2$ is such that $T_pM^1=T_pM^2$ and $N^1(p)=N^2(p)$.
Let $h^1,h^2$ be the local parameterizations by the tangent plane of $S^1$ and $S^2$ respectively, defined in the same neighbourhood of $p$ in the affine tangent plane $P$.
We say that $S^1$ is locally inside of $S^2$ in a neighbourhood $U\subset P$ of $p$ if we have that $h^1\geq h^2$.
I have the following questions:
- What does it mean to be a local parameterization by the tangent plane of $S^1$?
- In the definition of locally inside what does it mean for $h^1\geq h^2$?
From the definition of locally inside I would assume that $h^i$ would have to be a map into $\mathbb{R}$ but then it wouldn't be a parametrization of a surface nor a plane.
This problem comes from an exercises in my independent studies. Thanks in advance.
You're parametrizing the surfaces as graphs over the (common) tangent plane $T_pM^1=T_pM^2$. That is, in a neighborhood of $p$, both surfaces are parametrized by a neighborhood $U$ of the origin in $T_pM$, with $M^i$ being given locally as $\{(\vec u, h^i(\vec u)): \vec u\in U\}$. (Here we use a new coordinate system adapted to the geometry — namely, coordinates in $T_pM$ and a third coordinate orthogonal to the plane.)