Frame

Let $M$ be a smooth $m$-dimensional manifold. A frame on an open set $U\subseteq M$ is an ordered set of vector fields $\{v_1,\dots ,v_m\}$ such that for every $x\in U$ they constitute a basis of $T_{x}M$.

Every coordinate chart induces an associated frame, the coordinate frame.

The structure coefficients of the frame are defined by the commutation relations of the base vectors: $[X_i,X_j]=C_{ij}^k{X}_k$.

It has a corresponding dual concept: coframe on a manifold