I'm currently in a GR class and have come across the notion of parallel transport, and I've searched and searched the last few days to try and understand it but I just can't seem to wrap my head around it, so I'm hoping someone here can clarify for me. The way I picture parallel transport is thus; if I have a manifold M equipped with a metric, then I can construct a covariant derivative. I do this by using the Levi-Civita connection, which is characterised by having zero torsion and also preserving the metric. The upshot of this is simply that, a) the covariant derivative of the metric is zero, and b) because the connection is torsion free, then we have that the covariant derivatives w.r.t two different coordinates commute. Parallel transport is the act of, if you like, "picking up" a vector which is a member of the tangent space of a point p, and then moving it along come curve C whilst always keeping its orientation the same as it was at p. This is what confuses me; if we keep it "pointing in the same direction" by using the covariant derivative to vary its components in conjunction with the varying basis vectors, then surely at any given point along the line the vector will not generally be in the tangent space of that point. I was under the impression that it only makes sense to describe vectors as being in the tangent plane of the manifold at a point. The example I keep coming back to is that of parallel transport along a given latitude of a sphere (not a geodesic). I understand mathematically (i.e I can see why the equations show) that a vector will not be parallely transported along this line because it isn't a geodesic, however by naively looking at a sphere and imagining moving a vector around a fixed latitude I cannot see why the transported vector does not coincide with the original one. I understand also the argument of using a cone tangential to the circle, and this makes sense to me, but surely we should be able to arrive at that conclusion without resorting to the cone? So I guess my first question is this; when we parallel transport a vector along a curve, does the vector have to remain in the tangent space of each point, or does it have to stay parallel to itself?