Existence of Orthonormal Frames

Theorem
Let $\struct{M, g}$ be a $n$-dimensional Riemannian manifold with or without boundary.

Let $p \in M$ be a point.

Let $TM$ be the tangent bundle of $M$.

Let $U \subseteq M$ be an open subset.

Suppose $\tuple {X_j}$ is a smooth local frame for $TM$ over $U$.

Then for all $p \in M$ there is a smooth orthonormal frame $\tuple {E_j}$ over $U$ such that:


 * $\forall k \in \N : 1 \le k \le n : \map \span {\bigvalueat {E_1} p, \ldots, \bigvalueat {E_n} p} = \map \span {\bigvalueat {X_1} p, \ldots, \bigvalueat {X_n} p}$

In particular, for every $p \in M$ there is a smooth orthonormal frame $\tuple {E_j}$ defined on some neighborhood of $p$.