On a flat surface, can a holonomy can be nontrivial around certain curves

Question

On a flat surface, can a holonomy can be nontrivial around certain curves? How is this possible?

Answer 1

Try the cone $M=\{z=\sqrt{x^2+y^2}\}$ (missing its vertex $(0,0,0)$. Holonomy is nonzero around any parallel $z=c$. This happens because that curve does not bound a region in $M$.