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 $p$.

Proof

This theorem requires a proof. In particular: Use Gram-Schmidt on vectors $X_j$; obtain orthonormal vector fields $E_j$; denominators are nonvanishing so $E_j$ are smooth; apply this to any smooth local frame You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Definitions