Matt Hunt

Is thinking of a metric as a little bit like a generalisation of the idea of vector potential horribly wrong, or is this concept useful?

Think of the metric as a form of kinetic energy. You require a bilinear form to define the kinetic energy generally, this is the metric defined on the tangent space.

What you do is you get a ream of paper, take note how many sheets are in the ream and then measure the thickness and divide by the number of sheets.