Is this figure interpretable in terms of Holonomy Group and connections with different metrics?

Question

Given the following picture

I interpret the two loops as moving orthonormal frames along Levi-Civita connections.

In particular, based on the paragraph about parallel transport, I recognize that the transport in the green loop is given by the standard Euclidean metric $ds^{2}=dr^{2}+r^{2}d\theta ^{2}$ and in the blue loop by the standard form in polar coordinates, and thus preserves the vector $\partial \over \partial \theta$ tangent to the circle.

Therefore I conclude that the above figure is similar to the one for the Holonomy in Wikipedia and it shows that

Parallel transport on a sphere depends on the path. Transporting along

the blue loop - as opposite to the green loop -

yields a vector different from the initial vector. This failure to return to the initial vector is measured by the holonomy of the connection.

Is my analysis correct or at least acceptable? Or is there anything wrong that forces one to only accept a different homotopic (*) explanation about contractible curves, based only on the smooth structure but not at all on the metric of the manifold?

(*) Very minor, marginal note/question about the homotopic interpretation. In that case is obvious that the green loop is the constant loop and it is contractible but I doubt if you can rigorously affirm that the blue loop is $S^1$ generator, once you also admit at the same time that the loop can move on the sphere and more importantly that the frame can rotate: in the latter case it would be trivial to see a homotopic transformation of the phase from $[0,2\pi]$ to $[\pi,\pi]$, hence a constant, contractible loop again (maybe no, see my comment below)? Once more, my point is that the way the orthonormal frame rotates (while the point of the loop moves) would be debatable unless one specifies it through a connection and a metric tensor (in the name "orthonormal" itself a metric tensor is somehow assumed)

Answer 1

Now I see the point much better: the starting point is the observation of Baez that

The argument is global. Take a right-handed o.n. frame at any point of $S^2$. You can get any other right-handed o.n. frame at any other by applying some rotation in $SO(3)$, and this rotation is unique, so the space of right-handed o.n. frames is a copy of $SO(3)$.

Now we have to consider that to get the nontrivial loop generating $\pi_1(SO(3))$, you need to find a path in $SU(2)$ connecting 1 to -1 and then project that to $SO(3)$.

Then I recall that $SU(2)$ is isomorphic to the group of versors in the quaternions. Now, mathematically, Dirac's string trick is a visual demonstration (that corresponds to the original figure in the question as I'm going to explain) of the theorem that $SU(2)$ (which double-covers $SO(3)$) is simply connected. To say that $SU(2)$ double-covers $SO(3)$ essentially means that the unit quaternions represent the group of rotations twice over: in other words the blue loop is non contractible but "twice the blue loop" is contractible because it is a generator of the projective space.

Another good, visual explanation is that

$SO(3)$ is topologically a ball in 3D space, with antipodal points on the surface identified. $SU(2)$ is the 3-sphere: again a ball, but this time the whole border is to be thought of as a single point. Hopefully this helps you to see why they're locally similar, while $SU(2)$ is simply connected and $SO(3)$ is not.

This fully explains the doubts about the "homotopic interpretation" in question, while everything else about the connection and the metric can be confirmed.