Definition:Index Lowering
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Definition
Let $\struct {M, g}$ be a Riemannian manifold.
Let $\tuple {E_i}$ and $\tuple {\epsilon^i}$ be a smooth local frame and its dual coframe.
Let $g = g_{ij} \epsilon^i \epsilon^j$ be the local expression of $g$.
Let $X = X^i E_i$ be a smooth vector field.
Let $\hat g : TM \to T^* M$ be the Riemannian metric mapping.
Let $\map {\hat g} X = g_{ij} X^i \epsilon^j$ be the covector field.
Index lowering of $X$, denoted by $X^\flat$, is an isomorphism $\flat : TM \to T^*M$ such that:
- $X^\flat := \map {\hat g} X$
Also known as
$X^\flat$ is also called $X$ flat.
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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