A Can you give an example of a non-Levi Civita connection?

Joker93

Hello!!
Since connections in general do not require that we have a Riemannian manifold, but only a smooth manifold, I find it kind of weird that the only examples of connections that I find in the internet are those which use the Levi-Civita connection.
So, I wanted to know of any examples of non-Levi-Civita connections.
If somebody could also give an example of an easily-visualizable parallel transport (like on S^2) using that connection it would be great.

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lavinia

Gold Member
Hello!!
Since connections in general do not require that we have a Riemannian manifold, but only a smooth manifold, I find it kind of weird that the only examples of connections that I find in the internet are those which use the Levi-Civita connection.
So, I wanted to know of any examples of non-Levi-Civita connections.
If somebody could also give an example of an easily-visualizable parallel transport (like on S^2) using that connection it would be great.
Take a look at post #14 in

Joker93

That's great. But, upon reading your answer, I can't help but feel helpless in my effort of trying to find an intuitive way to think about it.

It also seems to me that parallel transporting a vector using Levi-Civita connection gives the closest thing there is to what we can intuitively call parallel vector. For instance, using the Levi Civita connection to parallel transport along the 2D Euclidean plane gives vector fields that are parallel with the usual everyday sense. And parallel transporting with the same connection along geodesics of a sphere gives a vector field that I can intuitively understand as what would an observer sitting at every point of that geodesic would perceive as parallel vector.

But, in the torsion-full examples that you gave in your answer, it seems to me that this intuition goes out of the window.
I am thinking that these other connections(that are not connected via variable changes) have to do with "moving observers"(moving frame of reference).

Would you please enlighten me a bit on how can I get some intuition for these torsion-full connections?
I have read about the difference about being torsion-full and torsion-less but I can't quite understand how can these parallel vector fields can be called parallel in any sense. Again, the only suspicion I have is viewing it as a moving(and rotating while moving) frame of reference.

Orodruin

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I suggest you read through all of that thread. In particular, the connection on the sphere (minus the poles) that I discussed has a very natural interpretation (a compass needle - or anything attached to it -- would be parallel transported) the connection preserves bearings and so the geodesics corresponds to curves of constant bearing.

Joker93

I suggest you read through all of that thread. In particular, the connection on the sphere (minus the poles) that I discussed has a very natural interpretation (a compass needle - or anything attached to it -- would be parallel transported) the connection preserves bearings and so the geodesics corresponds to curves of constant bearing.
Could you please help me define this way of parallel transporting?

Orodruin

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The connection is uniquely defined by defining its action on a complete set of basis vector fields. In this case, take the orthonormal basis fields in the coordinate directions and assume that they are paralell. The connection coefficients will follow.

Joker93

The connection is uniquely defined by defining its action on a complete set of basis vector fields. In this case, take the orthonormal basis fields in the coordinate directions and assume that they are paralell. The connection coefficients will follow.
How is this expressed mathematically though?

Orodruin

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As I said in the other thread, I am not going to write it down explicitly for the reason that it would be essentially verbatim copying an example from my upcomig textbook and I want to stay clear from doing so for contractual reasons. However, if you follow the steps I outlined above you should be able to reproduce it easily.

Joker93

As I said in the other thread, I am not going to write it down explicitly for the reason that it would be essentially verbatim copying an example from my upcomig textbook and I want to stay clear from doing so for contractual reasons. However, if you follow the steps I outlined above you should be able to reproduce it easily.
Do you know of any other source that might contain anything related to what I am asking?

fresh_42

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Do you know of any other source that might contain anything related to what I am asking?
Here's a nice lecture note about it:
http://www.uni-math.gwdg.de/amp/Markina3.pdf
By varying the number $1-6$ you get the other chapters.
I haven't read it all, yet, so maybe it's a bit too abstract and not specified enough, but it's a good starting point.

Joker93

Here's a nice lecture note about it:
http://www.uni-math.gwdg.de/amp/Markina3.pdf
By varying the number $1-6$ you get the other chapters.
I haven't read it all, yet, so maybe it's a bit too abstract and not specified enough, but it's a good starting point.
Thanks for the link, although I found that the notes have a very different presentation than that of my course(on just Riemannian geometry).
If you can think of something that is more close to Riemannian Geometry rather than bundles and stuff, please do post it here.
Thanks again

"Can you give an example of a non-Levi Civita connection?"

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