Distances with non-euclidean metric

[mnb96]

Hello,
when measuring length of geodesic shortest paths, or more in general, when measuring the length of a parametric curve in the space, what we usually do is to sum the length of infinitesimal arcs of that curve, assuming an euclidean norm.

Why this choice?
I have not found in literature any mention on the possibility of using other norms, like L1-norm.

Why not to allow to measure the length of infinitesimal arcs of a curve in $$\mathbb{R}^2$$ by doing instead:

$$ds = \left| \frac{\partial \mathbf{p}}{\partial x} \right| dx + \left| \frac{\partial \mathbf{p}}{\partial y} \right| dy$$

Thanks...

 

[arkajad]

We sum the lengths without any norm. We just sum numbers because of the natural assumption that the length of two pieces of a given path is the sum of the two lengths.

Here I am assuming that we are talking about the Riemannian and not pseudo-Riemannian case.

The other question is how to define the infinitesimal length. Here you have not only Riemannian but also more general Finsler geometries (see e.g. Chern and Shen, "Riemann-Finsler Geometry", World Scientific (2004)).