\begin{definition}[Definition:Index Lowering]
Let $\struct {M, g}$ be a Riemannian manifold.
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 $X = X^i E_i$ be a smooth vector field.
Let $\hat g : TM \to T^* M$ be the Riemannian metric mapping.
Let $\map {\hat g} X = g_{ij} X^i \epsilon^j$ be the covector field.
'''Index lowering of $X$''', denoted by $X^\flat$, is an isomorphism $\flat : TM \to T^*M$ such that:
:$X^\flat := \map {\hat g} X$
\end{definition}