Let $M$ be a smooth $m$-dimensional manifold. A frame on an open set $U\subseteq M$ is an ordered set of vector fields $\{v_1,\ldots,v_m\}$ 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
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Author of the notes: Antonio J. Pan-Collantes
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