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

• A
• Joker93
In summary, the conversation discusses the concept of connections on manifolds and specifically the Levi-Civita connection. The speaker is looking for examples of non-Levi-Civita connections and an easily-visualizable example of parallel transport using such a connection. They also mention difficulties with understanding the intuition behind torsion-full connections and ask for additional resources or explanations.
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.

Joker93 said:
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

lavinia said:
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.

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.

Orodruin said:
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.

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.

Orodruin said:
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?

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
Orodruin said:
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?

Joker93 said:
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.

fresh_42 said:
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

## 1. What is a non-Levi Civita connection?

A non-Levi Civita connection is a type of connection or covariant derivative used in differential geometry. It is a generalization of the more commonly known Levi Civita connection, which is used to define the notion of parallel transport on a manifold. Unlike the Levi Civita connection, a non-Levi Civita connection does not necessarily satisfy the properties of torsion and metric compatibility.

## 2. Why is it important to study non-Levi Civita connections?

Studying non-Levi Civita connections is important because it allows us to explore more general types of connections, which can be useful in various areas of mathematics and physics. For example, non-Levi Civita connections have been used in the study of higher-dimensional manifolds, such as Calabi-Yau manifolds in string theory.

## 3. Can you give an example of a non-Levi Civita connection?

One example of a non-Levi Civita connection is the Chern connection, which is used in the study of complex manifolds. It is defined in terms of the complex structure on the manifold and does not satisfy the properties of torsion and metric compatibility.

## 4. How are non-Levi Civita connections different from Levi Civita connections?

The main difference between non-Levi Civita connections and Levi Civita connections is that non-Levi Civita connections do not necessarily satisfy the properties of torsion and metric compatibility. This means that the parallel transport defined by a non-Levi Civita connection may differ from that defined by a Levi Civita connection.

## 5. In what areas of science are non-Levi Civita connections used?

Non-Levi Civita connections are used in various areas of mathematics and physics, including differential geometry, general relativity, and string theory. They have also been applied in other fields, such as computer graphics and machine learning, to improve the representation and manipulation of data on curved spaces.

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