Let \((x^i)\) be some local coordinates on a Riemannian manifold \((M,g)\). Then the inner product of two tensor fields \(F,G\in T^{(k,l)}TM \) is locally represented by

\begin{equation*} \langle F,G\rangle=g_{i_1r_1}\cdots g_{i_kr_k}g^{j_1s_1}\cdots g^{j_ls_l}F_{j_1\ldots j_l}^{i_1\ldots i_k}G_{s_1\ldots s_l}^{r_1\ldots r_k}. \end{equation*}

We obtain this result since the inner product is bilinear and by raising the index.

Remarks
  • Using convention of raising the index , we obtain for covariant tensor fields \begin{equation*} \langle F, G\rangle=F_{j_1\ldots j_l}G^{j_1\ldots j_l} \end{equation*} the last factor on the right represents the components of \(G\) with all its indices raised: \begin{equation*} G^{j_1\ldots j_l}=g^{j_1s_1}\cdots g^{j_ls_l}G_{s_1\ldots s_l}. \end{equation*}