Computing the first fundamental form

Question

Let $X$ be a smooth surface in $\mathbb{R}^{3}$. I want to compute the first fundamental form of $X$. Assume that $X$ has 2 different local parameterizations $r_1$, $r_2$ (i.e. for $r_1$ there is a $U_1 \subset X$ open and $V_1 \subset\mathbb{R}^{3}$ open such that $r_{1}: V_{1} \to U_{1}$ is a homeomorphism, $r_1$ is a smooth function and $\frac{d}{du}r_1(u,v)$ and $\frac{d}{dv}r_1(u,v)$ are linear independent. Analogously, we find $U_2 \subset X$ open and $V_2 \subset \mathbb{R}^{3}$ open, such that $r_2: V_{2} \to U_{2}$ satisfies the similar properties as $r_1: V_{1} \to U_{1}$. Moreover $U_{1} \cup U_{2}=X$.).

Assume there is a parameterization $f$ of $X$ in the general sense (i.e $f: U \to X$ surjective, where $U\subset \mathbb{R}^{3}$, but e.g. $f$ has not to be homeomorphism or has to satisfy other properties of the local parameterization above). Assume that $f$ is biijective.

Can I compute the first fundamental form with $f$ or do I have to compute a first fundamental form w.r.t. $r_1$ and $r_2$?

Answer 1

A surface $X$ is a subset of $\mathbb{R}^3$ locally parametrized by open sets in $\mathbb{R}^2$. That is, for each $q\in X$ there exists an open set $V$ in $X$ and an open set $U$ in $\mathbb{R}^2$ and a smooth function $$\vec r:U\to V \ \ \ \ \ \ \ \ \ \ \ \ \ \ (u,v)\mapsto \vec r(u,v)=(x(u,v),y(u,v),z(u,v))$$where $x,y,z:U\to \mathbb{R}$ are smooth functions. In your comment above, you're saying assume the surface is regular, i.e. the Jacobian of $\vec r$ has full rank, so that it's two columns, $\vec r_u,\vec r_v$ are linearly independent. Given the $q\in X$ above, there is $p\in U$ such that $\vec r(p)=q$. You can show that $\vec r_u(p),\vec r_v(p)$ is a basis for $T_pX$. Assuming we have a metric the first fundamental form is $$I:T_pM\to \mathbb{R} \ \ \ \ \ \ \ \ \ \ \ \ \ \ v\mapsto \langle v,v\rangle.$$ Suppose $v=a\vec r_u(p)+b\vec r_v(p)$. You can show that $$I(v)=Ea^2+2Fab+Gb^2$$ where $E=\langle\vec r_u(p),\vec r_u(p)\rangle, F=\langle \vec r_u(p),\vec r_v(p)\rangle\text{ and } G=\langle \vec r_v(p),\vec r_v(p)\rangle$.

So using the parametrization you can compute the first fundamental form. I'm not sure what your asking about the parametrizations $f$ and $\vec r_1$ or $\vec r_2$ is. You can use whatever regular parametrization you want.