Euclidean plane

From the point of view of classical differential geometry

The Euclidean plane is the manifold $M = R^{2}$ , with coordinates $(x_{1}, x_{2})$ , together with the natural Riemannian metric

g = d x_{1} \otimes d x_{1} + d x_{2} \otimes d x_{2} .

It is, therefore, a Riemannian manifold.

Seen as a plain manifold we can consider that $M$ is endowed with a natural linear connection $\nabla$ , such that $\nabla_{\partial_{x_{i}}} \partial_{x_{j}} = 0 \partial_{x_{1}} + 0 \partial_{x_{2}}$ . The induced principal connection on the frame bundle is given by the 1-form

ω = {(\begin{matrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end{matrix})}^{- 1} \cdot (\begin{matrix} d c_{11} & d c_{12} \\ d c_{21} & d c_{22} \end{matrix}) \in Ω^{1} (R^{2}, gl (2))

f = (\begin{matrix} c_{11} & c_{12} & x_{1} \\ c_{21} & c_{22} & x_{2} \\ 0 & 0 & 1 \end{matrix}) \in F M

Here it is shown how to construct $ω$ is constructed from an arbitrary $\nabla$ .

On the other hand, if we think of $M$ as a Riemannian manifold we can consider the Levi-Civita connection $\nabla_{L C}$ . Since the metric is constant, the covariant derivative $\nabla_{L C}$ coincides with the natural covariant derivative $\nabla$ , so it induces the same connection $ω$ on $F M$ .
But the metric $g$ also specifies a orthonormal frame bundle $O M$ (see here why). The elements of this principal bundle are

f = (\begin{matrix} c & - \sqrt{1 - c^{2}} & x_{1} \\ \sqrt{1 - c^{2}} & c & x_{2} \\ 0 & 0 & 1 \end{matrix}) \in O M

with $c \in [- 1, 1]$ .
Since

ω |_{O M} = (\begin{matrix} c & \sqrt{1 - c^{2}} \\ - \sqrt{1 - c^{2}} & c \end{matrix}) \cdot (\begin{matrix} d c & \frac{c}{\sqrt{1 - c^{2}}} d c \\ \frac{- c}{\sqrt{1 - c^{2}}} d c & d c \end{matrix}) =

= (\begin{matrix} 0 & \frac{d c}{\sqrt{1 - c^{2}}} \\ - \frac{d c}{\sqrt{1 - c^{2}}} & 0 \end{matrix}) \in Ω (R^{2}, o (2)),

according to Proposition 4.7. in vicenteBundles, this connection can be reduced to a connection on the orthonormal frame bundle determined by the metric $g$ .

If we parameterize this principal bundle $O M$ with

(x_{1}, x_{2}, θ) \mapsto (\begin{matrix} \cos (θ) & - \sin (θ) & x_{1} \\ \sin (θ) & \cos (θ) & x_{2} \\ 0 & 0 & 1 \end{matrix})

we obtain the more famous expression for $ω$ :

(\begin{matrix} 0 & d θ \\ - d θ & 0 \end{matrix})

Remember: this 1-form tells us how much the bases at $f$ and $f^{'}$ "fail to be constant" when we pass from the frame $f$ to a nearby frame $f^{'}$ , but expressing this mistake with respect to the frame $f$ itself.

From the point of view of Cartan geometry

The Euclidean plane is a Cartan geometry modeled over $(E (2), O (2))$ , indeed is the Klein geometry $(E (2), O (2))$ . Moreover, it is a reductive Klein geometry since

e (2) = {(\begin{matrix} C & v \\ 0 & 0 \end{matrix}) : C \in o (2), v \in R^{2}} = o (2) \oplus p

We have a natural choice for $p$

p = {(\begin{matrix} 0 & p \\ 0 & 0 \end{matrix}) : p \in R^{2}} .

With this in mind, remember that the Maurer-Cartan form describes all possible "infinitesimal displacements" of the frame $f$ , but from the point of view of the frame $f$ itself (see this). That is, if we pass from the frame $f$ to another frame $f^{'}$ , the Maurer-Cartan form at $f$ applied to the "vector" $\vec{f f^{'}} = (d x_{1}, d x_{2}, d θ)$ is a packet of information

A = (\begin{matrix} 0 & - d θ & c o s (θ) d x_{1} + s i n (θ) d x_{2} \\ d θ & 0 & - s i n (θ) d x_{1} + c o s (θ) d x_{2} \\ 0 & 0 & 0 \end{matrix}) \in e (2)

Here is encoded, on the one hand, how much have we moved the base point of $f$ to the base point of $f^{'}$ and, on the other, how much have we changed the basis itself. The natural choice of $p$ let us think that the information about the change of base point is in the $v$ part (the projection of the Maurer-Cartan form on $p$ ), and so the projection of the Maurer-Cartan form on $o (2)$ , $(\begin{matrix} 0 & d θ \\ - d θ & 0 \end{matrix})$ , tell us how much has the basis changed. That is, the same as the connection 1-form of the connection on the orthonormal bundle induced by the metric $g$ (which is the Levi-Civita connection).

To summarize:
In other words, in the orthonormal frame bundle induced by the metric $g$ we consider a displacement from a frame $f = (\begin{matrix} C & p \\ 0 & 1 \end{matrix})$ to a frame $f^{'} = (\begin{matrix} C^{'} & p^{'} \\ 0 & 1 \end{matrix})$ .
The principal connection $ω$ induced by the Levi-Civita connection measures the change from $C$ to $C^{'}$ as an element of $o (2)$ .
The Cartan connection (Maurer-Cartan form) measures the change from $f$ to $f^{'}$ as an element of $e (2)$ . This change can be decomposed like the union of a change from $C$ to $C^{'}$ and a change from $p$ to $p^{'}$ . This is reflected in the fact that $e (2) = o (2) \oplus p$ . If we focus on the change from $C$ to $C^{'}$ we have the principal connection $ω$ .