Which is to say that it applies to (smooth) manifolds (possibly with boundary); a polyhedron is not a smooth manifold! There are combinatorial proofs of the Gauss-Bonnet and umlaufsatz for polygons/polyhedra, but the smooth proofs I know use a smooth approximation to the pw-smooth polyhedra near the singularities.

And though it's certainly useful for physics, Riemann and Levi-Civita were certainly defining and working with curvature well before Einstein and his followers!

Another complex thing explained simply (rather like the... don't know the name for it... the thing in the top bit of my IE window... the bit that says "Sharp Blue: making the complex simple"... sort of the opposite of what I've just done with my aside... or rather tangent... well NOW it's a tangent... where was I?) well done Rich!

Just wondered whether you could insert "(corner)" at the end of the second sentence in the fourth paragraph. I was struggling for a bit because I mistook vertex for (whatever the mathmatical term for edge is), my maths vocabulary has been in decline for the last 9 years.

Barney

The thing at the top is the "title" in HTML speak, or the "tagline". I've also made the change you suggest.

DF said:

Which is to say that it applies to (smooth) manifolds (possibly with boundary); a polyhedron is not a smooth manifold! There are combinatorial proofs of the Gauss-Bonnet and umlaufsatz for polygons/polyhedra, but the smooth proofs I know use a smooth approximation to the pw-smooth polyhedra near the singularities.

I'm not sure I've ever seen a proof of the polyhedral version. I've skimmed Chern's proof of the theorem for closed, orientable Riemannian surfaces, but haven't worked through it in detail. Physicists have a shocking tendency to just accept theorems! And I, at least, tend to get intimidated by mathematicians when I learn things like the fact that the Gauss-Bonnet theorem is a special case of the Atiyah-Singer index theorem!

And though it's certainly useful for physics, Riemann and Levi-Civita were certainly defining and working with curvature well before Einstein and his followers!

Undoubtedly. Although Gauss tried measuring the curvature of space long, long before Einstein. It's also interesting that Einstein resisted Minkowski's attempt to geometrise special relativity, and didn't know anything about the work of Riemann et al when he formulated general relativity.

Similarly, physicists were working with theories of the other forces based on gauge symmetries, gauge-covariant derivatives and potentials long before they realised that such theories could be formulated as theories about connections on fibre bundles.

your stuff doesn't help!