Let $(M,g)$ be a Riemannian manifold, $M$ connected, $\nabla$ the Levi-Civita connection. The holonomy principle (see e.g. Besse "Einstein manifolds") states that there is a bijective correspondence between
- Tensor fields $t$ invariant under parallel transport.
- Parallel tensor fields, that is $\nabla t=0$.
- Tensors at $x\in M$ invariant under the holonomy group of $M$.
This entry of the Encyclopedia of Mathematics and my attempts at a proof seem to imply that 1. and 2. are always equivalent (1. $ \Rightarrow$ 2. is clear, I have some doubts about the reverse). Is that true?
So, suppose we replace holonomy with restricted holonomy in 3. I think 1. would become the statement that if $\gamma$, $\delta$ are homotopic curves then parallel trasport along $\gamma $ is equal to parallel transport along $\delta$. What happens to 2.?
EDIT I am getting more and more convinced that 2. does not imply 1. in the non-simply connected case. Consider $\pi: S^2 \to RP^2$ with the metric on $RP^2$ induced from the round metric on $S^2$. If $\gamma$ is a geodesic of $S^2$ such that $\pi\circ \gamma$ is non-contractible, then the tangent to $\pi\circ\gamma$ is a parallel vector field along $\pi \circ\gamma$, but parallel transport along $\pi\circ \gamma$ acts as minus the identity.
If anyone could provide a proof of the fact that vanishing covariant derivative implies that parallel transport is homotopy invariant, or whatever is the correct statement, and which does not use the Ambrose-Singer theorem, that would be great!