Let \(F\) be a \((k,l)\)-tensor field. Its \(i\)-th argument is a vector. Then the lowering of the \(i\)-th index is defined by
\begin{equation*} F^{\flat}(\alpha_1,\ldots,\alpha_{k+l})=F(\alpha_1,\ldots,\alpha_{i-1},\alpha^{\flat}_i,\alpha_{i+1},\ldots,\alpha_{k+l}) \end{equation*}Similarly, let \(G\) be a \((k,l)\)-tensor field and its \(i\)-th argument is a covector. Then the raising of the \(i\)-th index is defined by
\begin{equation*} G^{\sharp}(\alpha_1,\ldots,\alpha_{k+l})=G(\alpha_1,\ldots,\alpha_{i-1},\alpha^{\sharp}_i,\alpha_{i+1},\ldots,\alpha_{k+l}) \end{equation*}
Remarks
- \(F^{\flat}\) is a \((k-1,l+1)\)-tensor field.
- \(G^{\sharp}\) is a \((k+1,l-1)\)-tensor field.
- In order to obtain the components of \(F^{\flat}\) or \(G^{\sharp}\) in any local frame, we multiply by \(g_{kl}\) or \(g^{kl}\), respectively, and contract one of the indices of \(g^{kl}\) or \(g_{kl}\) with the \(i\)-th index.