Metrics As Mappings Between Arrows and Stacks

Another post for the Poincaré Project.

Back in Coordinate Systems and Metrics we saw that a metric for a coordinate system tells us the "distance travelled as proportion of coordinate change". Then in the following post, Metrics in Two or More Dimensions:

Imagine that you're at a particular point on a two-dimensional manifold. If you head off in a particular direction from that point at a particular rate, your coordinates will change. The metric tells you, from a given point, the rate of change of each of your coordinates given travel in a particular direction at a particular rate.

Those two posts express two sides of the same coin: in one I said the metric tells us the rate of change of position given the rate of change of coordinates and in the other I said the metric tells us the rate of change of coordinates given a rate of change of position.

A rate of change of position is, as we've seen, an arrow-vector. A rate of change of a particular coordinate is, as we've also seen, a stack-vector in the dual space.

In fact, one can view a metric as being a mapping between arrow-vectors and stack-vectors. You can use it, along with some calculus if the metric is different at different points, to calculate distances (as described in Coordinate Systems and Metrics). It can also be used to calculate the length of a vector or the angle between two vectors (concepts which don't exist without a metric).

A metric ties those length and angle notions to the coordinate system and, in so doing, actually defines the coordinate system.

Finally, a metric has within it, all the information necessary to describe the curvature of a manifold. It is ultimately this function that makes it relevant to both the General Theory of Relativity and the Poincaré Conjecture.

We will explore each of these in due course. The main takeaway at this point is that a metric is a mapping between arrow-vectors and stack-vectors.