Completion or extension of a frame

See @lee2013smooth proposition 8.11 (exercise)

Proposition

Let $M$ be a smooth $n$ -manifold with or without boundary and let $(X_{1}, \dots, X_{k})$ be a linearly independent $k$ -tuple of smooth vector fields on an open subset $U$ of $M$ , with $1 \leq k < n$ . Then for each $p \in U$ there exist smooth vector fields $X_{k + 1}, \dots, X_{n}$ in a neighborhood $V$ of $p$ such that $(X_{1}, \dots, X_{n})$ is a smooth local frame for $M$ on $U \cap V$ .

$◼$

In other words, we can complete to a frame.

I add that if the manifold $M$ is contractible space then the completion can be performed in the whole $M$ . Since in this case the tangent bundle is trivial,

g : T M \to M \times R^{n}

we can apply a linear transformation pointwisely

h : M \times R^{n} \to M \times R^{n}

such that $h \circ g$ sends $X_{i}$ to $(0, 0, 1, 0, \dots, 0)$ . The desired vector fields are given by the preimage (pre-pushforward, better said) of the rest of the canonical vector fields.

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es

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