You guys suck! My calculus class doesn't start until the fall so I can't currently appreciate what you've presented here. I hate being "out-nerded".

The Gauss-Bonnet article briefly touches on manifolds and reminded me about the other manifold - as in non-manifold geometry - which the Maya manual says is bad for subdivision surfaces. (rolls eye) Jeez Rep, you can be so one track-minded sometimes. And how exactly can you use manifolds to aid in deformations?

The tensor article did nothing for me which means that it probably contains information leading to my next revolutionary breakthrough.

I can only visualize differential geometry one of two ways - either as discreet points on a NURBS surface or as dpdu / dpdv functions used to anti-alias shaders in a Renderman-compliant shader builder.

While the possibility that a solid theoretical understand will create optimized meshes, I can't yet see the connection between the two. Once I can, well, the universe will be in serious trouble (again).