\[ \newcommand{\d}{\mathrm{d}} \newcommand{\e}{\mathrm{e}} \newcommand{\i}{\mathrm{i}} \]

Let \(\hat{g}^{-1}\) be the inverse of the tangent-cotangent isomorphism . Compared to the local coordinate representation of \(\hat{g}\), we obtain

\begin{equation*} \hat{g}^{-1}(\omega)(\eta)=g^{ij}\omega_j\frac{\partial }{\partial x^i}, \end{equation*}

where \(g^{ij}\) denotes the entries of the inverse of the matrix with the entries \(g_{ij}\).

It is customary to denote the components of \(\hat{g}(X)\) by

\begin{equation*} \hat{g}^{-1}(\omega)=\omega^i\frac{\partial }{\partial x^i}, \end{equation*}

where \(\omega^i=g^{ij}\omega_j\). This procedure is called raising an index.