Question

What is curvature?

Answer 1

Of a geometry (e.g. Euclidean curvature =0), and how do you quantify it?

Answer 2

Curvature is just a measure of the degree to which a line "turns."

Answer 3

I understand that, but I give you a random geometry. How can you tell me EXACTLY what the curvature of that geometry is? I think Riemann did it fairly simply, but I don't know how.

Answer 4

This is a little above my level, but I guess you have to measure the rate of convergence or divergence of parallel lines.

Answer 5

It's like this http://en.wikipedia.org/wiki/Gaussian_curvature (Which I don't really understand, could someone explain) generalised into higher dimensions.

Answer 6

He came up with a representation that is invariant under coordinate axis transformations by means of unit tangent vectors in terms of curve lengths. I will try to recall.

Answer 7

http://en.wikipedia.org/wiki/Curvature_of_Riemannian_manifolds

Or if I'm asking a stupidly hard question, the answer to which I won't understand, what do I have to know beforehand to understand it?

Answer 8

This is beyond the level of OpenStudy. I would recommend that you ask the same question here: http://math.stackexchange.com/questions/ask (Do not report me, guys!) There is a lot of knowledgeable people there in the area of differential geometry.

Answer 9

@henpen don't be shy - excellent explanation is here http://en.wikipedia.org/wiki/Curvature

Answer 10

Basically curvature is the normalized absolute value of the second derivative of a parametric curve equation.

Answer 11

What does normalised mean in this sense? Multiplied by a convenient constant?
Surely the absolute value of the second derivative of a parametric curve equation would depend on the equation. Do you mean the deviation from what the absolute value of the second derivative of a parametric curve equation would depend on the equation would be in Euclidean space?

Answer 12

Really sorry, I didn't understand much of that.

Answer 13

Yes it takes about two to three hours (at least.....) to explain well the vector differentiation and the curvature.

Answer 14

Correction: I understand loosely what the first half is saying, but the second half is still unfathomable.

Answer 15

Fair enough. How do I start learning differential geometry?

Answer 16

First do not kill your motivation : start with partial understanding here
http://www.youtube.com/watch?v=T143lv8aKDU
But before that may be it is better to understand here:
http://www.youtube.com/watch?v=2bJGctuQslI

Answer 17

Actually, I have a definition in a book somewhere. Thanks!

Answer 18

Hey @henpen why did you go ? All this time I have been looking FOR YOU where to start from.
Start from multivariable calculus
only THE diffrential geometry
Mathematics - Multivariable Calculus - Lecture 1
http://www.youtube.com/watch?v=cw6pHhjhKmk&feature=relmfu

Answer 19

Thank you very much for that, and sorry for my late reply. The outside world tugs at me occasionally, unawares that you are aiding me. Why did you choose UC Berkeley course over MIT?