Local Orthonormal Frame and Coframe related by Index Raising

Theorem
Let $\struct {M, g}$ be an $n$-dimensional Riemannian manifold.

Let $\tuple {E_i}$ be the local frame of $M$.

Let $\tuple {\epsilon^i}$ the local coframe dual to $\tuple {E_i}$.

Let $\sharp$ be the sharp operator.

Then the following are equivalent:


 * $\tuple {E_i}$ is orthonormal.
 * $\tuple {\epsilon^i}$ is orthonormal.
 * $\forall i \in \N_{1 \mathop \le i \mathop \le n} : \tuple {\epsilon^i}^\sharp = E_i$.