Definition:Riemannian Metric Mapping

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

Let $TM$ and $T^*M$ be tangent and cotangent bundles of $M$.

Let $p \in M$ be a base point.

Let $T_p M$ be the tangent space of $M$ at $p$.

Let $v, w \in T_p M$ be tangent vectors.

Then the bundle homomorphism is the mapping $\hat g : T M \to T^* M$ such that:

$\forall p \in M : \forall v, w \in T_p M : \map {\map {\hat g} v} w = \map {g_p} {v, w}$