The Levi-Civita connection commutes with musical isomorphisms , i.e. for some smooth tensor field \(F\) with a contravariant \(i\)-th index the lowering of the \(i\)-th index satisfy
\begin{equation*} \nabla(F^\flat)=(\nabla F)^\flat. \end{equation*}Similarly, if \(G\) has a covariant \(i\)-th index the raising of the \(i\)-th index satisfy
\begin{equation*} \nabla(G^\sharp)=(\nabla G)^{\sharp}. \end{equation*}
Remark
- Since the Levi-Civita connection also commutes with the trace operator , the same is true for the trace of covariant \(k\)-tensor fields with \(k\ge 2\), i.e. \begin{equation*} {\tr}_g (\nabla F)=\tr ((\nabla F)^\sharp)= \nabla \tr(F^\sharp)=\nabla {\tr}_g(F) . \end{equation*}.