Definition:Index Raising

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