Quantitative statement about torsion and rotating vectors

Question

In this MO answer, user 'anonymous' gives an example of a connection on $\mathbb{R}^3$, with torsion, where a vector parallel transported along a straight line will rotate uniformly on the plane perpendicular to that line. I would like to know if there are any general statements about an arbitrary 'torsionful' connection to the effect that parallel transported vectors must somehow rotate.

Answer 1

In the statement about $\mathbb{R}^3$, two connections are implicitly being used: the "torsionful" connection and the stanard Euclidean connection on $\mathbb{R}^3$. Indeed, it makes no sense to say that parallel transported vectors "rotate" unless they rotate relative to something. Still, a similar point can be made in general.

Let $\widetilde{\nabla}$ be an affine connection on a manifold $M$. We can describe the torsion of $\widetilde{\nabla}$ with its torsion tensor $\tau$, defined by $$ \tau(X,Y):=\widetilde{\nabla}_XY-\widetilde{\nabla}_YX-[X,Y] $$ One can show from this definition that $\tau$ is indeed a $(1,2)$ tensor, and that $\tau(X,Y)=-\tau(Y,X)$.

This tensor can be used to define a new connection $\nabla$ which is the "torsion free part" of $\widetilde{\nabla}$: $$ \nabla_XY:=\widetilde{\nabla}_XY-\tau(X,Y) $$ $\nabla$ is a torsion-free affine connection, and that the geodesics of $\nabla$ and $\widetilde{\nabla}$ agree.

Now, let $\gamma$ be a geodesic, and let $\widetilde{X}$ be a vector field along $\gamma$ which is parallel with respect to $\widetilde{\nabla}$. $$ \nabla_{\dot{\gamma}}\widetilde{X}=-\tau(\dot{\gamma},\widetilde{X}) $$ We can think of the partial evaluation $-\tau(\dot{\gamma}(t),\_)$ as an infinitesimal linear transformation which describes how $\widetilde{\nabla}$-parallel transport "twists" with respect to $\nabla$-parallel transport. One can describe this using a $\nabla$-parallel frame $(e_1,\cdots,e_{n-1},\dot{\gamma})$. $\widetilde{\nabla}$-parallel transport from $\gamma(t_1)$ to $\gamma(t_2)$ will give an $n\times n$ matrix which fixes the $\dot{\gamma}$ component, and $-\tau(\dot{\gamma}(t),\_)$ will be the infinitesimal generators of those matrices.

If $\widetilde{\nabla}$ is a metric connection, however, then $\nabla$ will be as well. In this case, $-\tau(\dot{\gamma}(t),\_)$ will be an infinitesimal rotation on the orthogonal complemet of $\dot{\gamma}(t)$, and the resulting parallel transports will differ by a rotation on $\dot{\gamma}^\perp$. This can be seen by working on an orthonormal frame.