\begin{definition}[Definition:Orthonormal Frame]
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'''.
\end{definition}