Doubt about of isometries representation in $\mathbb{R}^{3}$

Question

I reading the following Theorem in (Do carmo, Riemannian Geometry, Chaprter VIII, Section 5.)

Theorem 1. (Loiuville): Let $f:U \to \mathbb{R}^{n}$, $n\geq 3$, be a conformal trasnformation a open set $U\subset \mathbb{R}^{n}$. Then $f$ is the restriction to $U$ of a composition of isometries, dilations or inversions, at most one of each.

On the other hand I finded the following result:

Theorem 2. If $F$ is an isometry of $\mathbb{R}^{3}$, then there exist a unique translation $T$ and a unique orthogonal transformation $C$ such that $F=TC$.

Here, we have

Definition(Conformal map). $f:U\subset \mathbb{R}^{n}\to \mathbb{R}^{n}$ is conformal, if, and only if for all $p\in U$ and for all pairs of vectors $v_{1},v_{2} \in p$ is true that: $$\langle df_{p}(v_{1}), df_{p}(v_{2})\rangle\leq\lambda^{2}(p) \langle v_{1},v_{2}\rangle, \hspace{4mm}\lambda\neq0$$

and

Definition(Isometry). A a diffeomorphism $f: \mathbb{R}^{n}\to \mathbb{R}^{n}$ is isometry, if, and only if for all $p\in U$ and for all pairs of vectors $v_{1},v_{2} \in T_{p}\mathbb{R}^{n}$ is true that: $$\langle df_{p}(v_{1}), df_{p}(v_{2})\rangle = \langle v_{1},v_{2}\rangle_{p}, $$

My doubt are the following:

1. I think that as a particular case of conformal map we have the isometries, only put $\lambda(p)=1$, but I don't sure abuout this affirmation since in the conformal map we consider only vectors in $p$. Is true this affirmation?

2. If the affirmation (1) above is true, as From the Theorem 2. we have $F=TC$, and from Theorem 1. have other "face" for $f$. I understand that $C$ is an isometry, so $T$ form Theorem $2$ must be a inversion or a dilatation, but I belive that it isn't a dilatation, so it must be a inversion. does my idea work?

3. Why is the importance of the isometries? My professor talked about of congruence of trianglues, but I don't understand the importance of this reason. In other site I finded: More generally, Euclidean geometry can be defined as the totality of concepts that are preserved by isometries of Euclidean space, so I kept thinking in the importance of Isometries and this definition of Euclidean geoemtry.

Thanks in advance!

Answer 1

1. "Isometry" in theorem 1 of Liouville has to be taken a priori in the strict sense of your theorem 2 (i.e. an affine transformation preserving distances), not of your final definition, which is that of a local isometry.
2. $F=TC$ itself is an isometry. Viewed threw the prism of Liouville's theorem, it is the composition of an isometry ($F$), no inversion, and no dilation.
3. Your question is too wide for me but the explanation you found on the web for the importance of isometries is fine. See also this post.