Parallel Transport Along Geodesic

[diegogarcia]

TL;DR Summary

Simple question: Along a manifold geodesic curve are all vectors in the tangent plane transported in a parallel manner or is it just the tangent vector to the curve?

I am self-studying differential geometry.

Most text books on the subject are filled with abstruse symbols and very little actual examples. (Maybe most actual examples are intractable in a practical sense?)

But, thanks to computer algebra software (Maxima), I am making good progress by creating my own examples.

However, I am currently stuck on one point. To keep it simple I will limit things to 2-D surfaces in 3-D space.

Considering a geodesic curve on an arbitrary surface, the tangent vector to the curve is always parallel transported from point to point along the geodesic curve. But what about the other vectors in the tangent plane along the curve? Are they also parallel transported from point to point along the geodesic?

In the examples I have created I calculate that only the tangent vector to the curve has a covariant derivate equal to zero. I calculate that a general vector in the tangent plane along the curve has a covariant derivative that is not equal to zero.

From my interpretation of the abstruse textbooks all vectors in the tangent plane along the geodesic should have covariant derivatives that are equal to zero. Either my interpretation is wrong or my calculations are wrong.

Basically, I just need a "yes" or "no" answer to the above question. Then I can carry on with my study.

 

[jedishrfu]

Wikipedia has a topic on it

https://en.wikipedia.org/wiki/Parallel_transport?wprov=sfti1

 

[mathwonk]

I am sorry I don't know this subject but I can recommend an intuitively written book. chapter 5, I think. free download. I kind of think you can translate any vector, not just one tangent to the curve, but I don't know.
https://projecteuclid.org/ebooks/bo...oduction/toc/10.3792/euclid/9781429799843#toc

 

[jedishrfu]

In the wiki article they show a vector transport on a great circle triangle as an example and mention the ##/alpha# angle in relation to the area of the triangle.

 

[Ibix]

I believe it depends on your choice of connection and in general the answer is "no". However, if you use the metric compatible and torsion free connection (Levi-Civita, as we always do in General Relativity) it is "yes". Metric compatibility preserves the inner product, which tells you that vector moduli cannot change and nor can the angles between them. Torsion free prevents rigid rotation about the tangent vector, which is the only possible remaining change.

So GR texts may well treat the answer as "yes". In a more general differential geometry context I believe it's "not in general".

 

[lavinia]

In the examples I have created I calculate that only the tangent vector to the curve has a covariant derivate equal to zero. I calculate that a general vector in the tangent plane along the curve has a covariant derivative that is not equal to zero.

That is correct.

Comments: The key theorem to know is that starting with a vector at a point on a smooth curve there is exactly one way to parallel translate it along the curve to get a vector field whose covariant derivative is zero everywhere. This is true for any affine connection. It is a good, I would even say crucial, exercise to prove this for oneself.

If the connection is metric compatible then parallel transport preserves lengths of vectors and angles between parallel translated vectors. If the curve is a geodesic this means that angles to the geodesic are preserved. In the cases that you are studying, I assume that the connection is metric compatible. So any vector field along a geodesic that is not both of constant length and constant angle to the geodesic is not parallel.

 

[jbergman]

TL;DR Summary: Simple question: Along a manifold geodesic curve are all vectors in the tangent plane transported in a parallel manner or is it just the tangent vector to the curve?

I am self-studying differential geometry.

Most text books on the subject are filled with abstruse symbols and very little actual examples. (Maybe most actual examples are intractable in a practical sense?)

But, thanks to computer algebra software (Maxima), I am making good progress by creating my own examples.

However, I am currently stuck on one point. To keep it simple I will limit things to 2-D surfaces in 3-D space.

Considering a geodesic curve on an arbitrary surface, the tangent vector to the curve is always parallel transported from point to point along the geodesic curve. But what about the other vectors in the tangent plane along the curve? Are they also parallel transported from point to point along the geodesic?

In the examples I have created I calculate that only the tangent vector to the curve has a covariant derivate equal to zero. I calculate that a general vector in the tangent plane along the curve has a covariant derivative that is not equal to zero.

From my interpretation of the abstruse textbooks all vectors in the tangent plane along the geodesic should have covariant derivatives that are equal to zero. Either my interpretation is wrong or my calculations are wrong.

Basically, I just need a "yes" or "no" answer to the above question. Then I can carry on with my study.

Your description is making me think there is some confusion here and you are conflating several concepts.

So let's imagine a curve from point A to B. If the vector field of a parameterized curve is 0 at every point in the curve then it is a geodesic.

Now let's talk about parallel transport along our curve. If we start at A we can take any tangent vector in the tangent space at A and parallel transport along our curve. What that means is that we can construct a vector field along the curve such that the covariant derivative is derivative 0 at every point. So if we start with some arbitrary tangent vector at A then when we parallel transport it we get a different tangent vector at each point along the curve.

Another way to think about if we have a metric, is that we have our curve defining a geodesic and its tangent vector at rach point on the curve. Then we can compute the inner product between an arbitrary tangent vector at A with the geodesic's tangent vector at A and then parallel transport is just choosing a tangent vector at each point on the curve such that its inner product with the geodesic's tangent vector is the same as it was at A.

What thing that you are saying that doesn't quite make sense is to take the covariant derivative of a tangent vector at an arbitrary point on the curve. That is because the covariant derivative takes derivatives of vector fields not vectors. It needs to see how a vector changes in a neighborhood.

What some people sometimes do is slur this distinction and take derivatives of tangent vectors by assuming it is a constant vector field in some neighborhood. In that case, yes the covariant derivative obviously won't be 0 along the curve for each of these "tangent vectors".