How arbitrary are connections?

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In summary, the conversation discusses the process of defining an affine connection on a differentiable manifold without a metric and the level of arbitrariness involved in this choice. The only restrictions are that the connection must transform correctly under coordinate transformations and be invertible, but there may be additional restrictions due to global topology. The purpose of the connection is to define "parallel" transport, which can be defined in various ways. Some nice properties that can be asked for in a connection include being torsion-free, volume-preserving, and length-preserving.
  • #1

Matterwave

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Suppose that I have a differentiable manifold M on which I don't define a metric. I wish to define an affine connection on that manifold that will allow me to parallel transport vectors from one tangent space of that manifold to another tangent space.

How arbitrary can I make my choice?

I do know that connections must transform correctly under coordinate transformations in order to keep my vector still a vector if I transport it, but is that it? Are there other restrictions on how I can choose my connection?

For example, we see pictures of parallel transport on the two sphere and because we can see the 2 sphere embedded in 3-D space, we can "intuit" what parallel transport would be like on that sphere. However, am I confined to that choice? Could I define an affine connection on a two sphere (given no metric, so I don't have to worry about compatibility issues) that would parallel transport vectors completely unintuitively to how I "would" do it from my 3-D perspective? Could I make the vectors twist and turn in weird fashion?

It seems to me that the affine connection is quite arbitrary; however, I have also seen equations that link it to how basis vectors "twist and turn" as we move throughout the manifold, so I am confused on really how arbitrary it is.

Perhaps I am too reliant on bases? @_@
 
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  • #2
Generically speaking, a connection is a matrix-valued 1-form [itex]\omega^a{}_b[/itex] (actually, Lie algebra-valued). Then the covariant derivative of a vector field [itex]X = X^a \, e_b[/itex] is written

[tex]\nabla X = (d X^a + \omega^a{}_b X^b) \otimes e_a[/tex]

where [itex]e_a[/itex] is any frame (i.e., a differentiable choice of basis at every point in some open patch).

In principle the only restriction on the matrix [itex]\omega^a{}_b[/itex] is that it be invertible; i.e. [itex]\omega^a{}_b \in \mathfrak{gl}(n, \mathbb{R})[/itex]. There may be further restrictions due to global topology of the manifold, to insure that the connection is continuous everywhere.
 
  • #3
But, other than those pretty general restrictions, I am free to choose them however I want? In what sense is the connection giving me a "parallel' transport then, if it's so arbitrary?
 
  • #4
A connection gives you the definition of "parallel". The point is that you can define "parallel" however you want.

There are other nice properties you can ask for, such as

1. Torsion-free (i.e. vectors do not twist helically around paths)

2. Volume-preserving (i.e., having a symmetric Ricci tensor)

3. Length-preserving (i.e., metric compatibility)
 
  • #5
I see...thanks. =]
 

1. How do we determine if connections are arbitrary?

There is no set method for determining if connections are arbitrary. It largely depends on the context and the specific connections being examined. However, some factors that may indicate arbitrary connections include lack of a clear logical or functional reason for the connection, and the connection being based on personal or cultural beliefs rather than empirical evidence.

2. Can arbitrary connections have a negative impact?

Yes, arbitrary connections can have a negative impact. When connections are made without a solid basis or logical reasoning, they can lead to false conclusions and misguided actions. This can be harmful in various contexts, such as in scientific research or decision-making in businesses or governments.

3. Are all connections made by humans arbitrary?

No, not all connections made by humans are arbitrary. While some connections may be based on personal or cultural beliefs, others are based on empirical evidence and logical reasoning. It is important to critically examine the connections we make and strive for those that are well-supported.

4. How can we avoid making arbitrary connections?

To avoid making arbitrary connections, it is important to engage in critical thinking and seek evidence and logical reasoning to support our connections. It can also be helpful to seek diverse perspectives and consider alternative explanations before drawing connections.

5. What role do biases play in making arbitrary connections?

Biases can play a significant role in making arbitrary connections. Our personal beliefs and experiences can influence the connections we make, and we may be more likely to accept connections that align with our biases rather than critically evaluating them. It is important to be aware of our biases and actively work to mitigate their influence on our thinking.

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