# What does it mean to 'preserve the first fundamental form'?

I'm a bit confused about the phrase 'preserving the first fundamental form', or 'The Gaussian curvature is determined by the first fundamental form'.

For example, let's say I have two surfaces $M$ and $M'$. How can they possibly have 'the same fundamental form', when:

1. The first fundamental form is a concept defined relative to some parametrisation
2. The domains of the various fundamental forms will not even coincide: $T_pM\times T_pM$ vs $T_{p'}M'\times T_{p'}M'$.

I just don't see how to interpret the phrase 'having the same fundamental form' or something being 'determined by the fundamental form' given that it is such a 'relatively-defined' notion.

Any help in clearing this up would be much appreciated. Thanks

Oke here is a concrete example of something which confuses me:

This source defines the first fundamental form as follows:

$$I_P(U,V)=U\cdot V,\text{for } U,V\in T_PM\ \ (\subset \mathbb{R}^3)$$

From this definition it is clear that this is independent of parametrisation, since it is completely in terms of the inner product on $\mathbb{R}^3$. However, the confusion starts when they then give the following definition (These are notes by Theodore Shifrin):

Suppose $M$ and $M^*$ are surfaces. We say they are locally isometric if for each $P\in M$ there are a reg-param $x:U\to M$ with $x(u_0,v_0)=P$ and $x^*:U\to M^*$, with the property that $I_P=I_{P^*}^*$, whenever $P=x(u,v)$ and $P^*=x^*(u,v)$. That is, the function $f=x^*\circ x^{-1}$ is a one-to-one correspondence that preserves the first fundamental form.

Oke so I just don't see how this definition of a local isomertic surfaces is even well defined when working with the above definition of the first fundamental form. What does is possibly mean that $I_P=I_{P^*}^*$? Usually for a function equality they need to at least have the same domains, but that's not the case here.

The only thing I can imagine is that $f$ induces through it's derivative a map $T_PM\to T_{f(P)}M^*$, and so that we say that $$I_P=I_{P^*}^* \text{ iff } I_P(x,y)=I_{P^*}^*(f'(x),f'(y))$$

And so that in general for a map $f$ to preserve the first fundamental form, this above definition is what it means. At least this is independent of parametrisation, so that's nice.

• The first fundamental form is the inner product on the tangent space, not the matrix that represents it with respect to a basis, so that it is not «a concept defined relative to some parametrisation» Oct 11, 2015 at 20:13
• @MarianoSuárez-Alvarez Yes I found that out, that seemed to have been my first source of confusion. There's more though, unfortunately. Oct 11, 2015 at 20:14

• Thanks for your answer. Unfortunately I do not really follow, since I have only studied the more 'concrete' case of surfaces in $\mathbb{R}^3$ so far. Would you say there is value in reading up on the more general theory to get a better view of what is going on here? Oct 11, 2015 at 18:18
• There is always value in reading up more general theory :-) But it's reasonable to go for that value only if it fits into your plans. The important point here is that, while the first fundamental form is defined using a parametrization, it is a concept which does not depend on the parametriziation. Roughly speaking, if you have a 2 surface $M$ in $\mathbb{R}^3$ and look at the tangent space $T_pM$ for some $p\in M$, the first fundamental form ist the restriction of the ambient scalar product to $T_pM$. This does not depend on a parametrization, only it's local representation does. Oct 11, 2015 at 18:38
• Oke, but if you don't mind me asking a more detailed question: I get that the function $I_p$ does not depend on the parametrisation, since this is, as you described, fairly easy to see. However, the proof of the Theorema Egregium expresses the curvature in terms of $E,F,G$ and their derivatives. These values DO depend on the parametrisation. So I find it confusing to speak as this depending only on $I_p$, while in fact these values also depend on the parametrisation... Oct 11, 2015 at 18:42