Definition:Trace of Tensor

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Let $\struct {M, g}$ be a Riemannian manifold.

Let $h$ be a covariant $k$-tensor field with $k \ge 2$.

Let $h^\sharp$ be a $\tuple {1, k - 1}$-tensor field obtained from $h$ by raising its index.

Then the trace of $h$ with respect to $g$ is a covariant $\paren {k - 2}$-tensor field defined as:

$\tr_g h := \map \tr {h^\sharp}$

where $\tr$ is the trace over a covariant and a contravariant index.