Suppose I have a curved space-time with metric $g_{\mu \nu}$ and with the connection coefficients $\Gamma^\alpha_{\mu \nu}$. Given some vector $v$, the expression for parallel transport of this vector along a curve $x^\mu(\sigma)$ is

$$ \frac{d v^\beta}{d\sigma} + \Gamma^\beta_{\mu \nu} v^\mu \frac{d x^\nu}{d \sigma} = 0. $$

Now, I want to consider parallel transport of this vector around an infinitesimal closed parallelogram spanned by the vectors $a$ and $b$ and study the change in $v$ upon performing this procedure. Suppose we start with vector $v$ at point $P$, specified by the initial condition

$$ v_P^\beta \equiv v^\beta(\sigma_P). $$

Then, the change in $v$ upon parallel transport along the closed infinitesimal loop is given by

$$ \Delta v = v^\beta_{|| P} - v_P^\beta = -\left(R^\beta_{\lambda \nu \mu} \right)_P v_P^\lambda a^\nu b^\mu $$

where the Riemann curvature tensor is $$ R^\beta_{\lambda \nu \mu} = \frac{d \Gamma^\beta_{\lambda \mu}}{d x^\nu} - \frac{d \Gamma^\beta_{\lambda \nu}}{d x^\mu} + \Gamma^\beta_{\alpha \nu} \Gamma^\alpha_{\lambda \mu} - \Gamma^\beta_{\alpha \mu} \Gamma^\alpha_{\lambda \nu} $$

My question is the following: suppose that the vector $v_P$ is orthogonal to the infinitesimal loop originally i.e., $$ v_P \cdot a =0,\quad v_P \cdot b = 0. $$ Then, is there a general class of metrics for which $\Delta v = 0$?

On playing around with this problem, I found that if the Riemann curvature tensor is maximally symmetric, then this is certainly true but I'm curious if there's a broader class of metrics for which this is true. For simplicity, we can even consider working in just 3 spatial dimensions and not worry about 3+1 space time dimensions to begin with. I'd love to know if there's some general statements made about when parallel transport of a vector along a loop orthogonal to the vector's initial orientation leaves the vector invariant.

  • 3
    $\begingroup$ duplicate of mathoverflow.net/q/291866 $\endgroup$ – Ben Crowell Jan 31 '18 at 23:34
  • $\begingroup$ (I see this has been answered on the duplicate above.) The intuitive picture for Einstein-Cartan gravity (gravity with torsion) is that parallel transporting along $a$ then $b$ is different to transporting along $b$ then $a$. $\endgroup$ – Colin MacLaurin Feb 7 '18 at 13:35
  • $\begingroup$ @ColinMacLaurin The question hasn't actually been answered in the presence of torsion, so in case you have some insight regarding that case, it would be of tremendous help! Does my equation for $\Delta v$ need modification in the presence of torsion? And what would the constraints on the manifold be in that case? $\endgroup$ – Aegon Feb 11 '18 at 6:17
  • $\begingroup$ Sorry but at present I haven't studied torsion, and was just quoting from a recent conversation with an expert. $\endgroup$ – Colin MacLaurin Feb 12 '18 at 2:19

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