Parallel transport around a loop

In summary, the Riemann curvature tensor measures the failure of parallel transport to return a vector to its original position in the tangent space. This failure is dependent on the commuting of vector fields X and Y, and the presence of torsion in the system. In the absence of torsion, the quadrilateral formed by the vectors will close, and the vector Z will return to its original position. This is due to the fact that X and Y commute, which can be proven using the torsion tensor.
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From the Wikipedia article on the Riemann curvature tensor:
Suppose that X and Y are a pair of commuting vector fields. Each of these fields generates a pair of one-parameter groups of diffeomorphisms in a neighborhood of x0. Denote by τtX and τtY, respectively, the parallel transports along the flows of X and Y for time t. Parallel transport of a vector Z ∈ Tx0M around the quadrilateral with sides tY, sX, −tY, −sX is given by
[tex]\tau_{sX}^{-1}\tau_{tY}^{-1}\tau_{sX}\tau_{tY}Z.[/tex]
This measures the failure of parallel transport to return Z to its original position in the tangent space Tx0M.

The last sentence assumes that ##\tau_{sX}^{-1}\tau_{tY}^{-1}\tau_{sX}\tau_{tY}## returns Z back to x0. It isn't obvious to me that this should be true. My guess is that this is true because X and Y commute, but I can't think of a proof for this claim.
 
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  • #2
X and Y being commuting fields is equivalent to the quadrilateral being closed (provided there is no torsion).

In the presence of torsion, quadrilaterals of commuting vectors fail to close, and the torsion tensor measures the amount by which they fail to close:

[tex]T(X,Y) \equiv \nabla_X Y - \nabla_Y X - [X,Y][/tex]
Here ##Z = T(X,Y)## is the vector that closes the gap left after taking into account the possibility that X and Y fail to commute.
 
  • #3
X and Y being commuting fields is equivalent to the quadrilateral being closed (provided there is no torsion).

That's what I suspected. Do you have a proof for this?
 

1. What is parallel transport around a loop?

Parallel transport around a loop is the process of moving an object along a closed path while keeping it parallel to itself at all times.

2. Why is parallel transport around a loop important in science?

Parallel transport around a loop is important in science because it allows us to analyze and understand how objects move in curved spaces, such as in General Relativity or in fluid dynamics.

3. How is parallel transport around a loop different from regular transport?

In regular transport, an object is moved from one point to another along a specified path. In parallel transport around a loop, the object is moved along a closed path while maintaining its orientation.

4. What is the mathematical concept behind parallel transport around a loop?

The mathematical concept behind parallel transport around a loop is called covariant differentiation. It involves defining a connection between tangent spaces at different points along the loop and using this connection to move the object along the loop while maintaining its parallelism.

5. How is parallel transport around a loop used in real-world applications?

Parallel transport around a loop has many applications in physics and engineering. For example, it is used in calculating the curvature of spacetime in General Relativity, in analyzing fluid flow around curved surfaces, and in computer graphics to create smooth animations.

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