Exploring Curves and Torsion in 3D and 4D: A Comparison

[TrickyDicky]

In 3D the torsion  measures how rapidly the curve twists out of the osculating plane in which it finds itself momentarily trapped.
So in 4D, would torsion measure how rapidly a curve twists out of the osculating 3-hypersurface in which it finds itself momentarily trapped? Or torsion of a curve does not generalizes this way in a 4 dimensional setting?

 

[Ben Niehoff]

After thinking about this, I think you are exactly correct.

Think about it this way:

A tangent line is a first-order approximation to the curve, and depends on one derivative.

The curvature gives the second-order approximation that depends on the second derivative and measures how the curve bends away from its tangent.

The torsion gives the third-order approximation that depends on three derivatives and measures how the curve bends away from the plane defined by its curvature and tangent.

So in 4-space, there would be a 4th-order approximation that depends on four derivatives and measures how the curve bends away from the hyperplane defined by its tangent, curvature, and torsion.

And so on and so forth.

 

[Ben Niehoff]

In fact, Wikipedia has an article on curves in arbitrary dimensions: http://en.wikipedia.org/wiki/Differential_geometry_of_curves

 

[TrickyDicky]

After thinking about this, I think you are exactly correct.

Think about it this way:

A tangent line is a first-order approximation to the curve, and depends on one derivative.

The curvature gives the second-order approximation that depends on the second derivative and measures how the curve bends away from its tangent.

The torsion gives the third-order approximation that depends on three derivatives and measures how the curve bends away from the plane defined by its curvature and tangent.

Ok, this is the easy part.

So in 4-space, there would be a 4th-order approximation that depends on four derivatives and measures how the curve bends away from the hyperplane defined by its tangent, curvature, and torsion.

And so on and so forth.

Does this give us two torsions, the third-order and the 4th-order approximations?
Or when talking about higher dimensional spaces only the last order is called torsion?

 

[Ben Niehoff]

I think in dimensions higher than 3, they don't really have names. The Wiki article calls them "generalized curvatures" and gives them numbers. If it were me, I'd probably always call the 2nd-order one "curvature" and the 3rd-order one "torsion"...then I might call the rest "hypertorsion", I dunno.

The names don't really jibe with the words used in intrinsic differential geometry. The curvature tensor measures 2nd-order effects, but the torsion tensor actually measures 1st-order effects! I'm not sure if 3rd- and higher-order effects can be measured locally, intrinsically speaking (or else in Riemannian geometry we'd have a whole tower of curvature tensors defined using nth derivatives of the metric).

 

[TrickyDicky]

Thanks Ben, that hint about terminology helps.

In the intrinsic case of a torsion tensor in n-dimensions I think it is about how the whole tangent n-space twists around a curve in the n-manifold so it makes intuitive sense that it measures first-order effects and depends on the connection (first derivatives).