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

483
30
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.
Thanks in advance!
 

lavinia

Science Advisor
Gold Member
3,071
533
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.
Thanks in advance!
Take a look at post #14 in

https://www.physicsforums.com/threads/arbitrariness-of-connection-and-arrow-on-sphere.922779/
 
483
30
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.

Thanks in advance.
 

Orodruin

Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
2018 Award
15,932
5,910
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.
 
483
30
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

Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
2018 Award
15,932
5,910
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.
 
483
30
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

Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
2018 Award
15,932
5,910
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.
 
483
30
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

Mentor
Insights Author
2018 Award
11,561
8,015
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.
 
483
30
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
 

Want to reply to this thread?

"Can you give an example of a non-Levi Civita connection?" You must log in or register to reply here.

Related Threads for: Can you give an example of a non-Levi Civita connection?

Replies
4
Views
3K
  • Posted
Replies
2
Views
3K
Replies
1
Views
2K
Replies
5
Views
940
Replies
7
Views
4K
Replies
3
Views
4K
  • Posted
Replies
1
Views
749

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top