Understanding Intrinsic and Extrinsic Curvature: An Intuitive Explanation

[Gza]

Could someone give me an intuitive example of extrinsic and intrinsic curvature. That would be much appreciated, thanks in advance.

 

[adityatatu]

hello,
I am not too sure but maybe the following link will help...
http://www.phys.utb.edu/~warren/courses/Phys5393/lecture16.pdf

 

[Gza]

Yes, that was very helpful. I had to dig up a softer book on a treatment of tensors, but it still served its purpose, thanks again.

 

[pmb_phy]

Yes, that was very helpful. I had to dig up a softer book on a treatment of tensors, but it still served its purpose, thanks again.

The cylinder is an excellant example of zero curvature. It is also an excellant example of a manifold for which there are infinitely many geodesics between any two points on the surface.

Pete

 

[Gza]

The cylinder is an excellant example of zero curvature. It is also an excellant example of a manifold for which there are infinitely many geodesics between any two points on the surface.

Maybe I'm referring to the wrong concept, but I thought a circle had a curvature inverse of its radius, so wouldn't the curved part of the cylinder have curvature?

 

[pmb_phy]

Maybe I'm referring to the wrong concept, but I thought a circle had a curvature inverse of its radius, so wouldn't the curved part of the cylinder have curvature?

You're referring to a different kind of curvature. In the case of the cylinder - when someone says that the surface has zero curvature they mean that there is no "intrinsic" curvature. However it does have an "extrinsic" curvature.

Pete

 

[mathwonk]

As I recall, again from reading Spivak some 35 years ago, Gaussian curvature of a surface at a point p, may be defined as the product of the curvature of the two (perpendicular) curves through p having respectively maximum and minimum curvature as curves. So for a cylinder, you are right that the curve of maximal curvature through the point is a circle of positive curvature, but the curve through the point with minimum curvature is a line with curvature zero, so the product, the curvature of the surface, is zero. Intuitively this is true because the cylinder can be flattened out without tearing it, so really it is not curved as a surface.


I do not know what intrinsic and extrinsic curvature mean but i can guess. Curvatiure is determined by a way of emasuring lengths i.e. a "metric". If a surface like a doughnut for instance is embedded in three space then there are many ways to define a length on it. There is the "extrinsic length" which is just the restriction to the doughnut of the notion of euclidean length. The associated curvatuire would be the extrinsic curvature. E.g. it was extrinsic curvature we were discussing above for the cylinder.

But it seems intuitively clear to me that we could define length differently, in a such a way that the length on (the surface of) a doughnut agreed with the extrinsic length on a cylinder and then the curvature of a doughnut surface would be zero.

So really all curvature is intrinsic, since it is determined by the metric, but you may call the metric and the associated curvature extrinsic if ity happens to agree with that of the embedding space. This is just a plausible guess, but not an uninformed one.