Context: I work on a scientific computing code (mostly fluid mechanics) and I am trying to learn differential geometry, understand it and then mimic its basic structures into a numerical code.
My intuition is that the "mesh" used in numerical simulations is nothing but the discrete version of the manifold.
I understood that a manifold is a topological space and an atlas (at least for continuum mechanics applications).
I think about a CW Complex as the topological discretization of the mesh. It is a finite set of cell spaces (i.e., points, lines, surfaces and volumes) related to each other by means of its boundary maps (i.e., the incidence matrices). This can be coded easily.
Question: If the CW Complex captures the topological nature of the manifold, which entity captures its "position"? (may we call this the "geometric nature" of the manifold?)
I guess I should "discretize the atlas" but, is there any structure that already does that? Is there something I am missing?
Thank you! :)
If you build a mesh, then the different line segments also each have a length. A CW complex wouldn't know about that because that is part of the geometry. The CW complex only describes which line segments are attached to which nodes. You could make some line segments very short and others much longer. The object will look very different because the geometry is very different but the topology didn't change.