# 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 bundle homomorphism.

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**.

## Sources

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