Let suppose I have a Riemannian manifold and I define a connection on it. Is it true and safe to say that if a connection is torsionless and I parallel transport a vector along a loop made by geodesics I should have the same vector at the end of the loop?
Can anybody clarify this to me in plain english (I will translate in a formal way by myself). Thanks in advance.
Edit. I asked this question because I wanted to have a clearer picture about Torsion. So let's suppose for a while to have a Manifold with zero curvature. Is then safe to say that if the connection is torsionless when I make a loop along geodesics then the angle of the transported vector is the same as the original one?
This is not true if the curvature is not zero. Every Riemannian connection is torsionless but it is not always flat as shows $S^2$ endowed with the metric inheritated from $R^3$. The Lie algebra of holonomy around loops is charcterized by Ambrose Singer which expressed it with the curvature, and a curvature is not always zero in a Riemannian manifod.
https://en.wikipedia.org/wiki/Holonomy#Ambrose.E2.80.93Singer_theorem