I’m not very familiar with differential geometry and am coming from a general relativity background, so would appreciate help with a question from that context. If this question could be posed in a more abstract sense, that would also be of tremendous help!

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.