= Musical isomorphism
{title2=$\flat_g,\sharp_g$}
{wiki}
= Metric duality between tangent and cotangent bundles
{synonym}
A <Riemannian metric> gives inverse <vector bundle isomorphisms>
$$
\flat_g:TM\to T^*M,\quad v\mapsto g(v,\cdot),\qquad
\sharp_g:T^*M\to TM.
$$
In coordinates they have <matrices> $g_{ij}$ and $g^{ij}$. Every <smooth manifold> admits a <Riemannian metric> by a <partition of unity>, so its tangent and <cotangent bundles> are isomorphic as real smooth <vector bundles>. This isomorphism depends on the metric, not merely on the manifold.
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