My vague understanding of parallel transport is that when the guassian curvature is zero for all points in an area enclosed by a loop then a vector moved around that loop remains unchanged as long as the area is simply connected and smooth. When a vector is transported in a loop around a cone it is shifted by an angle even though the gaussian curvature is $0$ throughout, this is attributed to the pointy (non smooth) part of the cone which happens to be enclosed in the loop. Say we remove the top part of the cone, we will see that the vector still shifts by the deficit angle around the loop, this is explained by the hole enclosed by the loop which says the manifold is not simply connected so my understanding from above doesn't apply. So why does the cylinder not shift the angle of the vector even though the loop encloses atleast one hole. The cone and cylinder can both be laid flat into a sheet of paper but the cone shifts the vector whereas the cylinder doesn't. What is the mathematical attribute of the cylinder which the cone doesn't posess that lets it be more like a sheet of paper than the cone. Im sure I understand it intutively but I'd like some concrete mathematical concept or theorem that I can look up and go deeper. Sorry if this is too vague and unclear. I'm learning this from non-rigourous physics lectures. Ask questions if you want me to be more clear. I'll edit the question
edit: since a sawed of cone and cylinder have the same gaussian curvature throughout and both are smooth and neither is simply connected but the vector that moves around a loop in the sawed off cone shifts whereas the cylinder doesnt, so what is the mathematical property that distinguishes the intrinsic curvature of the cylinder and the cone