Simplicial curvature & simplicial Gauss-Bonnet

Introduction and text

The last section of Bloch’s chapter 3 (simplicial surfaces) is a long (and to my mind at this time, uninteresting) proof of the 2D Brouwer fixed point theorem: any continuous map from the disk to the disk has a fixed point. Bloch also proves a corollary, the no-retraction theorem, that there is no continuous map r from the disk to the circle such that r(x) = x for all x on the circle.

That one sounds interesting. We’ve seen in before, with the commentary that you can’t map the surface of a drum onto its rim without tearing it. I still don’t see it that way. But it is rather shocking that the map r cannot preserve all the points on the rim.

Anyway, we’re not going to fight with those. For me, the climax of chapter 3 is the simplicial Gauss-Bonnet theorem. It shows that there is a definition of curvature for simplicial surfaces (in fact, for polyhedra in general) such that the total curvature of a surface is equal to times its Euler characteristic .

(A simplicial surface is a polyhedron all of whose faces are triangles. I expect we’ll see this again in another post.)

That the total Gaussian curvature of a surface is equal to is called the Gauss-Bonnet theorem. It is a reasonable culmination of a first course in differential geometry. The simplicial version means that we have a definition of curvature for simplicial surfaces and polyhedra which gives us a form of the Gauss-Bonnet theorem. That says it’s a reasonable definition of curvature.

So what is this marvelous definition of simplicial curvature? It’s also called the angle defect, and goes back to Descartes.

The Euler Characteristic: Teasers

These are things i came across when I first started looking at the Euler characteristic, in fact, when I was looking at triangulations in particular.

n-manifolds

The Euler characteristic generalizes to dimensions other than 2, and there are at least three noteworthy theorems involving the Euler characteristic. I’m not going to say much about them, because they, like so much else, are still outside my comfort zone. I’ll just barely tell you what they are, and leave you to chase them down if they interest you.

As we’ve seen, the Euler characteristic of a polyhedron is given by

,

where v, e, f are the numbers of vertices, edges, and faces. Homeomorphic polyhedra have the same Euler characteristic, and that means we can define the Euler characteristic of a topological surface as the Euler characteristic of any polyhedron which is homeomorphic to it.

This alternating-sign sum of the numbers of 0-, 1-, and 2- simplices generalizes in the obvious way: for an n-simplex, we take the sum, with alternating signs, of the numbers of k-simplices, for k <= n. As for surfaces, so for n-manifolds: this is a topological invariant, and we want to define the Euler characteristic of an n-manifold as the Euler characteristic of any k-simplex homeomorphic to it.