Why Does Parallel Transport on a Circle Cause Vector Field Rotation?

[murmillo]

I'm trying to understand parallel transport and I'm stuck. The example given is if you have a unit sphere and you take one of the latitudes (not the equator), take at a point on the latitude the tangent vector to the curve, and parallel transport it around the curve. I don't understand why the vector field rotates counterclockwise. According to the notes, "If we just take the covariant derivative of the tangent vector to the circle, it points upwards, so the vector field must rotate counterclockwise to counteract that effect in order to remain parallel." I don't understand that sentence. Is it supposed to be obvious that the covariant derivative of the tangent vector points upwards? And what does he mean by counteracting the effect? Any help would be appreciated--I've spent a long time trying to understand this.

 

[Kreizhn]

Hey,

Not sure if this will help, but here's my two cents. Keep in mind this is in no way a rigorous explanation, but I definitely found that it helped me.

I assume you're using the Levi-Cevita connection for a covariant derivative? I always had trouble seeing how the tangent vector moved the way it did under parallel transport, until I read a nice little statement in Lee's book on Riemannian Geometry.

Essentially, imagine that instead of moving the tangent vector along a curve, you instead "move the entire manifold" along the curve in such a way as keep the tangent vector oriented in the same direction. Naturally in order to visualize this you have specified some sort of embedding, which won't always work. However, in the case of the unit sphere there is a natural embedding into \mathbb R^3. When you translate this back into transporting the tangent vector, you'll see it behaves in precisely the same way as if you had moved the manifold.

Now it's hard to reflect on what you mean by "counterclockwise" since this is not well defined. Furthermore, even given an embedding of the unit sphere in \mathbb R^3 and a fixed observation point, whether the vector field moves clockwise or counterclockwise is dependent the direction one travels on the curve. Since you haven't specified the curve or the direction of transport, it's hard to say anything more than that.