Let \((x^i)\) be some local coordinates on a Riemannian manifold \((M,g)\). Then the inner product of covector fields \(\omega, \eta\) is locally represented by
\begin{equation*} \langle \omega, \eta\rangle=g^{ij}\omega_i\eta_j. \end{equation*}
Remarks
- Using convention of raising the index , we obtain \begin{equation*} \langle \omega, \eta\rangle=\omega_i\eta^i=\omega^j\eta_j. \end{equation*}