6

After viewing a lecture on torsion, the lecturer said that the torsion is the failure of curves to close.
Since this is almost also what I have read about the Lie bracket, I want to know their difference and also understand it geometrically.

The Lie bracket is included in the definition of torsion, so I am guessing that the "non-closure" of curves due to torsion has to do with non-zero Lie bracket.
1) But, what if we have a zero Lie bracket and non-zero torsion? Does thit mean that the curves will again fail to close?
2) And what if we have a zero torsion and a non-zero Lie bracket? Does this mean that the curves will again fail to close? If not, in what way, graphically, can we say that the covariant derivatives found in the definition of torsion compensate for the effect of the Lie bracket?

EDIT:
For completeness, I also give the definition of the torsion tensor:
$T(X,Y):=\nabla_XY-\nabla_YX-[X,Y]$

TheQuantumMan
  • 2,478

1 Answers1

4

In simple words(not formal): The torsion describes how the tangent space twisted when it is parallel transported along a geodesic. The Lie bracket of two vectors measures, as you said, the failure to close the flow lines of these vectors.
The main difference is that torsion uses parallel transport whereas Lie bracket uses flow line.
This image is not mine. I saved it from MSE months ago
enter image description here

Semsem
  • 7,651
  • The Lie bracket is included in the definition of the torsion. So, the flow line information is also encoded in the torsion. Also, are you sure that the torsion has to do with parallel transport only along geodesics? I thought that it's about parallel transport along any curve. Lastly, when some people try to explain torsion, they refer to the twisting of coordinate frames(as in moving coordinate axis) as we perform parallel transport along curves. How does your explanation tie into that? Thank you. – TheQuantumMan Nov 01 '17 at 21:59
  • Thank for you the picture, this is absolutely fantastic! – Siddharth Bhat Aug 15 '19 at 20:15