Let \(\hat{g}\) be the tangent-cotangent isomorphism . Then due to the local coordinate representation of \(g\) we may write
\begin{equation*} \hat{g}(X)(Y)=g_{ij}X^iY^j. \end{equation*}Therefore the representation of \(\hat{g}(X)\) in local coordinates is given by
\begin{equation*} \hat{g}(X)=g_{ij}X^idx^j. \end{equation*}It is customary to denote the components of \(\hat{g}(X)\) by
\begin{equation*} \hat{g}(X)=X_jdx^j, \end{equation*}where \(X_j=g_{ij}X^i\). This procedure is called lowering an index.