3 - Metric structures

Summary

In this chapter we introduce a metric on our open subset of ℝⁿ. For the moment let us consider a surface that can be parametrized by two coordinates and thereby be viewed as a subset of ℝ². We should not imagine it flat – nor curved. An adequate model is a rubber sheet where lengths and angles are undefined. What the surface inherits from ℝ² is its topology, those properties which do not change when the sheet is smoothly stretched or otherwise deformed. Typical properties are: dimensionality, connectedness, holes. Note that the dimensionality of a hole has no meaning. For example if we cut a point, ‘zero-dimensional hole’, out of our sheet and then stretch it, the hole can become one- or two-dimensional.

Technically, a deformation is given by a diffeomorphism F (active transformation). A vector field can be thought of as little arrows drawn on the rubber sheet. Deformations naturally act on vector fields by displacing the base points of the arrows and by turning and stretching or shrinking them. This action was named tangent mapping T_xF in chapter 2.

A metric fixes lengths and angles on that rubber sheet. It becomes rigid, but not in the sense of being plastered, because this would imply an embedding of the sheet in our three-dimensional space, thereby also fixing angles and lengths in directions normal to the surface.