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.

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

Proof

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Sources

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