Cartan geometry ed

new version of Eichfeldtheorie der Cartanbasen

useful knowledge:

• Klein geometry
• Fiber bundles
• Lie groups
• Differential forms

Cartan structure ed

Definition ed

First we fix a model geometry, i.e.

• lie algebras \[\mathfrak h \subset \mathfrak g\] ("infinitesimal Klein geometry")
• Lie group \[H\] of \[\mathfrak h\]
• a representation \[Ad : H \rightarrow GL(\mathfrak g)\] that extends \[Ad : H \rightarrow GL(\mathfrak h)\]

Cartan gauge \[\theta\]

a \[\mathfrak g\]-valued 1-form \[\theta\] on \[U \subset M\] so that

\[ T_uM \stackrel{\theta}{\rightarrow} \mathfrak g \rightarrow \mathfrak g / \mathfrak h \]

is an isomorphism (right side is the canonical projection)

(here \[\omega_H\] is the Maurer-Cartan form of \[H\])

Cartan atlas

• \[M\] covered by Cartan gauges \[(U,\theta_U)\]
• on overlap there is \[ k : U \cap V \rightarrow H \] with \[\theta_V = Ad(k^{-1}) \theta_U + k^\star \omega_H \]

Curvature

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

Interpretation ed

This is equivalent to the usual moving frame language by using the Euclidean group \[G=\operatorname{Euc}_n\] and \[H=SO_n\]. The gauge \[\theta\] can be expanded into rotational and translational parts. The Rotational parts are then the usual connection forms \[{\omega^i}_j\] while the translational parts are the frames \[\theta^i\].

The above definition of curvature splits into the two structure equations with the usual curvature \[{\Omega^i}_j\] and torsion \[\mathcal T^i\] being the rotational and translational parts of the new curvature \[\Theta\].