Moving frame

Remarks

A moving frame on $U$ is the same as a smooth section over $U$ of the frame bundle of $M$. Example: on $\R^2 \setminus \set{0}$ with polar coordinates, $E_1 = \partial_r$ and $E_2 = \frac1r \partial_\theta$ form a moving frame; at each point they are orthonormal for the Euclidean metric, so they form a basis. Non-example: on $\R^2$, the pair $(\partial_x, x \, \partial_y)$ is not a moving frame, because at the points with $x = 0$ the second field vanishes. Given an affine connection $\nabla$, the connection forms $\omega^i_j$ of the frame are defined by $\nabla E_j = \sum_i \omega^i_j \otimes E_i$, and Cartan's structure equations express torsion and curvature through $d\theta^i$ and $d\omega^i_j$. A moving frame on all of $M$ exists only when the tangent bundle is trivial; for example $S^2$ has none.