Cartan geometry ed
new version of Eichfeldtheorie der Cartanbasen
useful knowledge:
Table of Contents
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
- 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
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\[ \Theta_r + \Theta_t = d\theta_r + d\theta_t + \frac{1}{2} \underbrace{[\theta_r \wedge \theta_r]}_{\in \mathfrak h} + \frac{1}{2} \underbrace{[\theta_t \wedge \theta_t]}_{=0} + \underbrace{[\theta_r \wedge \theta_t]}_{\in \mathfrak g_t} \]which splits into the two structure equations\[ d\theta_t = - [\omega \wedge \theta_t] + \mathcal T \]\[ d\omega = - \frac{1}{2} [\omega \wedge \omega] + \Omega \]when identifying \(\theta_t\) with the usual frame \(\theta^i\), \(\theta_r\) with the connection \({\omega^i}_j\) and \(\Theta_{t,r}\) with the torsion \(\mathcal T\) and curvature \(\Omega\).