Question

What is the curvature of a geometry, and how can you quantify it?

Answer 1

You can define the curvature of a manifold extrinsically, i.e. by embedding it in a higher dimensional space and then considering the radii of tangent circles, or intrinsically by considering the change of orientation of a vector when after moving it via parallel transport in a closed loop. What is the context of the curvature you're trying to understand?

Answer 2

Not in any physical context, if that's what you're implying.

Answer 3

No, I mean what makes you ask? In what context did you happen upon then term "curvature" and want to understand it better?

Answer 4

In a popularisation talking about how Riemann generalised Gauss' 2D curvature theorem. http://en.wikipedia.org/wiki/Curvature_of_Riemannian_manifolds , in essence.

Answer 5

I understand that much like the earlier eigenvalue problems that my prior knowledge isn't sufficient, but here (that is, differential geometry) I don't even know where to start.

Answer 6

Differential geometry is a discipline that even most mathematicians don't reach until they're in graduate school. I don't want to put you off but I am obligated to recommend that you become exceedingly comfortable with calculus and linear algebra first.

Answer 7

No, that's better than false motivation, I suppose. Will work on that first then (I think this has been the resolution to your last answer to my question also, I've obviously been putting it off to an extent with tangents). Thank you verily, regardless.

Answer 8

Sure. Keep up the good work.

Answer 9

Sorry, I wasn't compliment fishing. I assume you would have to know DG to really understand curvature.