The following is an excerpt from Pressley's Elementary Differential Geometry on the definition of the first fundamental form. However, there are some parts of this concept that I'm unclear about. It says in the bottom of this excerpt that the coefficients E,F,G and the linear maps du, dv depend on the choice of surface patch for $S$. Here, the patch is $\sigma$. But then, I don't understand how the first fundamental form is an inherent concept of the surface, that is the expression varies under different parametrizations of the same surface and the same point on it. So I can't understand why in the final line, it says "but the first fundamental form itself depends only on $S$ and $\mathbf{p}$.

So to make my question clear, there are theorems that show that the fundamental form determines many properties of different surfaces, such as surfaces that have the same fundamental form are locally isometric etc. But from what I understood from the below definition is that the first fundamental form is an expression, of which is determined by the coefficients $E, F, G$ and the linear maps $du, dv$. But these depend on the parametrization of the surface. So for the same surface and the same point on it, we can have two different expressions,say, $Edu^2+2Fdudv+Gdv^2$ and $\bar{E}d\tilde{u}^2+2\bar{F}d\tilde{u}d\tilde{v}+\bar{G}d\tilde{v}^2$. So the specific choice of parametrization seems to matter, but then how can we use this to talk about inherent properties of different surfaces? I would greatly appreciate any help to understand this.

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If $T$ is a two dimensional vector space, an euclidian structure is given by a symmetric bilinear form, positive, non degenerate. In a given coordinate system $(x,y)$ it is given by $ Ex^2+2Fxy+Gy^2$. $E,F,G$ depends on the coordinate system, but the euclidian structure do not.

This is what happen in your case, just you have a two parameter family of vector spaces $T_p$ (the tangent space at the point $p$ on $S$), and on each space an euclidian structure : if $S\subset E_3$ is a surface in the euclidian space, it is the restriction of the euclidian structure to the tangent space at this point.

If you choose a chart $(u,v)$ around your point $\partial u, \partial v$ is at each point a base of the tangent space, and in this base (which depends on the point) the euclidian structure at the point $p$ of coordinate $(u,v)$ is $g(\partial u,\partial u)=E(u,v),g(\partial u,\partial v)=F(u,v)$...The functions depends on the coordiante system, the euclidian structure not.

  • 1
    $\begingroup$ What do you mean by a Euclidean structure? $\endgroup$ Feb 20 '20 at 2:09
  • 2
    $\begingroup$ Exactly this a postiive definite quadratic form. If you prefer a distance which satisfies euclidian axioms. $\endgroup$
    – Thomas
    Feb 21 '20 at 4:47

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