Given a Riemannian manifold \((M,g)\) and a covariant \(k\)-tensor field \(T\) in \(M\) with \(k\ge 2\). Then raising of the last index results into a \((1,k-1)\)-tensor field. We set the trace of \(T\) as
\begin{equation*} {\tr}_g T = \tr(T^\sharp), \end{equation*}where the trace is taken over the last two indices.
The trace of \(T\) is a covariant \((k-2)\)-tensor field.
In local coordinates the coordinates of \({\tr}_g T\) is given by
\[ ({\tr}_g T)_{i_1\cdots i_{k-2}}= {T_{i_1\cdots i_{k-2}m}}^{m}=g^{ml}T_{i_1\cdots i_{k-2}ml}. \]