I have seen the phrase "The metric allows for a canonical identification of the tangent space with the cotangent space" all over diff geo resources and questions. I understand the map and why it serves as an identification, but since it works using an inner product, isn't the inner product what allows for the identification? I understand that inner products can induce metrics, but I don't see how the metric comes into play here.
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The "metric" is another name for the inner product. In differential geometry, the term "metric" usually refers to a Riemannian metric, that is a smoothly varying family of inner products on the tangent space at each point (or, if you are looking at just one tangent space at a time, "metric" can simply refer to the inner product on that tangent space). This is distinct from the term "metric" as used for metric spaces (though it is of course related, since a Riemannian metric induces a metric space structure on a connected manifold via the geodesic distance function).
Eric Wofsey
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