Klein geometry ed

A generalization of Euclidean geometry and other highly symmetric spaces.

required Fiber bundles and Lie groups

Table of Contents

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

Categories: Mathematik, Differentialgeometrie