# Definition:Index Raising

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## Definition

Let $\struct {M, g}$ be a Riemannian manifold.

Let $\hat g : TM \to T^* M$ be the Riemannian metric mapping.

Let $\tuple {E_i}$ and $\tuple {\epsilon^i}$ be a smooth local frame and its dual coframe.

Let $g = g_{ij} \epsilon^i \epsilon^j$ be the local expression of $g$.

Let $g^{ij}$ be the inverse matrix of $g_{ij}$.

Let $\omega$ be a smooth covector field.

Let $\map {\hat g^{-1}} \omega = \omega_i g^{ij} E_j$ be the vector field.

**Index raising of $\omega$**, denoted by $\omega^\sharp$, is an isomorphism $\sharp : T^*M \to TM$ such that:

- $\omega^\sharp := \map {\hat g^{-1}} \omega$

## Also known as

$X^\sharp$ is also called **$X$ sharp**.

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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