\begin{definition}[Definition:Index Raising]
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:
:$\omega^\sharp := \map {\hat g^{-1}} \omega$
\end{definition}