[andrewla]

In differential geometry the metric [1] is a tensor that defines the relationship of vectors in the space to vectors in the tangent space. The identity function as a metric means that you are in a locally flat space where geodesics (the path taken by traveling in a given direction) are straight lines.

A metric in a traditional metric space is a global distance function; you can use the metric tensor in a Riemannian manifold to allow integration to find the distance between two points.

[1] https://en.wikipedia.org/wiki/Metric_tensor

[throwawaymath]

Ah, so that's the setting we're talking about. Thanks for explaining that.