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

Question

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

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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.

Answer 1

First fundamental form is the traditional name of the Riemannian metric in a Riemannian manifold of dimension two, i.e. the scalar product on the tangent bundle. This is independent of a coordinate system but can be expressed (and is usually defined) using a coordindate system. A map preserving the first fundamental form is just a (local) isometry of Riemannian manifolds.

Answer 2

This sort of question is very common in differential geometry. One does wonder if tangent plane at a point P of a surface M is dependent on the parametrization at that point. Such arguments are generally dealt with introduction of a function called change of parameters or change of coordinates.

You could look up section 2.3 of Curves and Surfaces, by Sebastian Montiel and Antonio Ros for a discussion on change of parameters. There is result in that section that proves that any change of parameters is a diffeomorphism. By having this result, we can see that the structures don't change irrespective of parametrization that we choose to work with.