Definition:Trace of Tensor
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Definition
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.
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Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Basic Constructions on Riemannian Manifolds