 Connection 1-form

Connection one-forms are defined by $\nabla e_b={\omega }^a{}_b\otimes e_a$. For the Levi-Civita connection they obey Cartan's torsion-free equation $d{\theta }^a+{\omega }^a{}_b\wedge {\theta }^b=0$ and metric compatibility ${\omega }_{ab}=-{\omega }_{ba}$.