A Simple Example of a Connection With Torsion

While learning general relativity as an undergrad, I was uneasy with the idea of torsion of an affine connection. Familiar phrases like “torsion measures how a frame twists as it undergoes parallel transport” seemed too lofty to serve as a helpful mental model.

The following example is one which gave me an aha! moment.

A “rolling” connection on .

Consider a connection on the plane defined in such a way that a vector undergoing parallel transport rotates in the -plane in proportion to its motion along the -direction. That is, define a connection so that the vector field defined by

which spins counter-clockwise when moving in the -direction, is covariantly constant, .

We can write the covariant derivative

more easily by reading the Latin indices as the two components of a matrix. Then, for differentiation in the -direction, we have

where Gamma_x arrow(u) is matrix multiplication. Enforcing implies that the matrix must be

The -direction is trivial, since does not depend on . And so,

is zero only when

This connection has zero curvature…

If we calculated the Riemann curvature

we would find that this connection is flat, since (all Christoffel symbols are constant) and

also vanishes since .

…but non-zero torsion

However, the torsion

does not vanish:

We can see that the torsion measures the wavenumber controlling the rate at which vectors spin when parallel transported along the axis.