Why do mathematicians study curvature and torsion?

[Tenshou]

I don't really understand the point in Curvature and Torsion, I am wondering if someone could explain them to me. Thank you for your kindness: Why do mathematicians need Curvature and Torsion? What are their main uses??

 

[yenchin]

You should post this under Differential Geometry. Anyway, a quick answer is: Curvature describes how "curve" is a curve, more precisely it tells you how much it deviates away from being a straight line [which has zero curvature]. Does that help? The torsion of a curve tells you how much it deviates away from a plane.

 

[Tenshou]

Yes, Tremendously ! Thank you~

 

[WannabeNewton]

EDIT: Ok I removed my previous response because for some reason the title now says theory of curves, which I did not see before for some reason. The concept of the torsion and curvature of a curve embedded in $\mathbb{R}^{3}$ are very easy to picture and define and can be found in any text on the classical theory of curves and surfaces. See Do Carmo "Differential Geometry of Curves and Surfaces" for an extended discussion about torsion and curvature that is particularly intuitive; this will expand upon everything yenchin said. The important point to take home is that for space curves, torsion and curvature uniquely specify the curve up to a transformation under an element of the euclidean group.

 

[yenchin]

Since the OP has in its title "Understand The Theory of Curves", I guess he was just referring to the "classical theory" of curves embedded in $\Bbb{R}^3$. If one uses the intrinsic definition of curvature [i.e. Riemann curvature], this always gives zero curvature for curve.

By the way to expand a little bit on my previous post: If the curvature is zero at every point, then the curve is a straight line; if the torsion is zero at every point, then the curve is planar [can be drawn on a plane; a helix is an example of non-planar curve].

Also curvature and torsion uniquely species any smooth curve [Fundamental Theorem of Curves].

What is interesting to me is that using curvature and torsion one can construct the Frenet-Serret frame, which also generalizes to higher dimensions [you would then have a bunch of "hypertorsions", which describes deviation away from hyperplanes of various dimensions].

 

[yenchin]

By the way, in the physics literature, one could generalize curvature and torsion and hypertorsions to timelike curves in Minkowski spacetime of special relativity. These describe worldlines of particles/observers. It is well known that accelerated observer in Minkowski spacetime does not see vacuum but a heat bath, an effect called Unruh Effect. This is because accelerated observers see different quantum vacuum than the stationary/constant-speed observer. However it is not as well-known that, depending on the different behaviors of the curves [which depends on its curvature, torsion and hypertorsions], the spectra that one sees are actually different. See John R. Letaw "Stationary world lines and the vacuum excitation of noninertial detectors" in Phys. Rev. D 23, 1709–1714 (1981). Furthermore, in the paper "http://jmp.aip.org/resource/1/jmapaq/v15/i6/p774_s1?isAuthorized=no" by Eli Honig et al., they relate the curvature, torsion and hypertorsions to combinations of the electric and magnetic fields.