Klein geometry ed

A generalization of Euclidean geometry and other highly symmetric spaces.

required Fiber bundles and Lie groups

Abstract definition ed

Klein geometry

A principal fiber \[H\]-bundle \[(G,H)\] with closed \[H \subset G\], yielding a connected space \[M = G / H\].

Kernel \[K\]

Largest subgroup \[K \subset H\] that is normal in \[G\].

If \[K=1\], we call \[(G,H)\] effective. If \[K\] is discrete, we call it locally effective. Can replace \[(G,H)\] by its associated effective geometry \[(G/K, H/K)\].

Intuition ed

When starting from a space \[M\], the group \[G\] captures the symmetries of some structures. \[H\] is the subgroup that keeps a single point fixed. The quotient of both groups gives back the set of points.

Example: Euclidean space ed

Looking at the group of all Euclidean motions (preserving distance and angles):

\[ G = E(n) = T(n) \rtimes O(n) \simeq \mathbb R^n \rtimes O(n) \]

Stabilizer of the origin is

\[ H \simeq O(n) \]

And the plane is

\[ M = G / H \simeq \mathbb R^n \]

Example: sphere ed

The sphere \[ S^n \subset \mathbb R^{n+1} \] is symmetric under rotations and reflections:

\[ G = O(n+1) \]

An arbitrary point on the sphere is fixed under the subset of rotations/reflections:

\[ H \simeq O(n) \]

The result is

\[ S^n \simeq O(n+1) / O(n) \]

Cartan gauge ed

(\[U \subset M\])

Definition ed

Gauge

A gauge is just a section \[ \sigma : U \rightarrow G\].

This is equivalent to a bundle chart \[ \psi : U \times H \rightarrow \pi^{-1}(U) \], because \[\sigma(u) := \psi(u, e) \] defines a gauge and also \[\sigma\] fixes \[\psi\] completely by \[\psi(u,h) = \sigma(u) \cdot h\].

Infinitesimal gauge

This is the pullback of the Maurer-Cartan form \[\omega_G\] through a gauge:

\[ \theta = \sigma^{\star} (\omega_G) \]

It automatically fulfils the structure equations

\[ d\theta + \frac{1}{2} [\theta \wedge \theta] = 0 \]

It is also the Darboux derivative of \[\sigma\]. In some sense, it is \[\sigma\] locally, but forgetting the identity element in its image group.

For \[ \sigma_2 = \sigma_1 \cdot h \], we get

Gauge transformation

\[ \theta_2 = \operatorname{Ad}(h^{-1}) \theta_1 + h^\star \omega_H \]

Example: Euclidean plane ed

Euclidean group \[ G = \operatorname{Eucl}(2) = \left\{ \begin{pmatrix} R & t \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} \cos \phi & \sin \phi & x \\ -\sin \phi & \cos \phi & y \\ 0 & 0 & 1 \end{pmatrix}, x,y,\phi \right\} \].

The origin is fixed by the subgroup \[ H = \left\{ \begin{pmatrix} R & 0 \\ 0 & 1 \end{pmatrix} \right\} \simeq SO(2) \]

G's Lie algebra has three generators \[ X, Y, \Phi \], with

\[[X,Y] = 0, \quad [X, \Phi] = Y, \quad [Y, \Phi] = -X \]

The Maurer-Cartan form of \[G\] is (by magic):

\[ \omega_G = \Phi \, d\phi + \cos \phi (X \, dx + Y \, dy) + \sin \phi (X \, dy - Y \, dx) \]

A gauge is defined by picking an angle \[\phi(x,y)\] at every point in the plane \[M\]. The pullback for an arbitrary \[\phi(x,y)\] creates the infinitesimal gauge

\[ \theta = (\cos \phi X - \sin \phi Y + \frac{\partial \phi}{\partial x} \Phi) dx + (\cos \phi Y + \sin \phi X + \frac{\partial \phi}{\partial y} \Phi) dy \]

A straight forward calculation verifies, that \[ d\theta + \frac{1}{2} [\theta \wedge \theta] = 0\].