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 \[G=\operatorname{Eucl}(n)\] and \[H=SO(n)\]. The algebra \[\mathfrak g\] splits into the rotational part \[\mathfrak h\] and a translational part equivalent to \[\mathfrak g/\mathfrak h \simeq T_uM\].

This also splits the gauge \[\theta\] into rotational and translational parts \[\theta = \theta_r + \theta_t\]. The Rotational parts are then the usual connection forms \[{\omega^i}_j\] while the translational parts are the frames \[\theta^i\].

Inserting this into the above definition of the curvature gives

\[ \Theta_r + \Theta_t = d\theta_r + d\theta_t + \frac{1}{2} \underbrace{[\theta_r \wedge \theta_r]}_{\in \mathfrak h} + \frac{1}{2} \underbrace{[\theta_t \wedge \theta_t]}_{=0} + \underbrace{[\theta_r \wedge \theta_t]}_{\in \mathfrak g_t} \]

\[ d\theta_t = - [\omega \wedge \theta_t] + \mathcal T \]

\[ d\omega = - \frac{1}{2} [\omega \wedge \omega] + \Omega \]