Definition:Orthonormal Frame

Definition
Let $M$ be a Riemannian manifold.

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

Let $p \in M$ be a point.

Let $T_p M$ be a tangent space of $M$ at $p$.

Let $\tuple {E_i}$ be the local frame for $M$ on $U$.

Suppose for each $p \in U$ the vectors $\bigvalueat {E_i} p$ are an orthonormal basis for $T_p M$.

Then $\tuple {E_i}$ is said to be an orthonormal frame.