## Thought Experiment

Consider a 2-sphere, $S^2$, and let $p$ be a point at the equator.

**Case 1**

Let us parallel transport a vector, $V$ from $p$ using the recipe:

- Move one unit of length East.
- Move one unit of length North.
- Move one unit of length West.
Move one unit of length South.

**Case 2**

Let us parallel transport a vector, $V$ from $p$ using the recipe:

- Move one unit of arc East.
- Move one unit of arc North.
- Move one unit of arc West.
- Move one unit of arc South.

## Differences

It's clear that the first case draw a path which is not closed, while the second path closed on itself. And moreover, the vectors might change, in the usual way, under the action of an element of the Holonomy group.

**Questions**

- Why does that difference appear?
- Am I misunderstanding the notion of parallel transport? (In that case which one is the correct one)
- It is well known that one might choose different connections in order to define the parallel transport. Is the above difference due to a choice of connection?
- Are these point of view the corresponding to Levi-Civita and Weitzenboeck connections?

I'd thank any help to clarify my doubts. Cheers.