- #1

diegogarcia

- 16

- 3

- 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.

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.