Antisymmetric connection (Torsion Tensor)

In summary, an antisymmetric connection, also known as the torsion tensor, is a mathematical concept used in differential geometry to describe the underlying geometry of a manifold. It is a measure of the failure of a connection to be symmetric, meaning that the order of differentiation does not matter. Antisymmetric connections play a crucial role in the study of Riemannian and pseudo-Riemannian manifolds, where they help define the geometry and curvature of the space. The torsion tensor can also be used to define the concept of parallel transport, which is essential in understanding the curvature of a manifold and the behavior of geodesics.
  • #1
psimeson
19
0
How to show:

Tabc = [itex]\Gamma[/itex]abc - [itex]\Gamma[/itex]acb

is a Tensor of rank (1,2)

Attempted solution:
1. Using definition of Covariant Derivative:
DbTa= ∂aTa+[itex]\Gamma[/itex]abcTc (1)

DcTa= ∂cTa+[itex]\Gamma[/itex]acbTb (2)

I subtracted (2) from (1) but I couldn't really get a Tensor out of it. I just got lost in the mess.

Is this is the right way to start it?
 
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  • #2
Do you have to use covariant derivatives in your problem? Is it a hint in your problem? There are several ways to show your property.

And why do you say "antisymmetric connection?"
 
  • #3
I am also in the process of learning tensor calculus, so I may not be right, but wouldn't it work if you raised the indices and made every tensor ab-contravariant?
 
  • #4
Which text are you using? There are different ways of showing your property, but the method should be adapted to what you already know.
 
  • #5
@arkajad: Covariant derivative is not a hint in the problem. I am just trying to solve that way. I am following various kind of textbooks. So, any way would work for me.

@meldraft: I am sure if that will work. Since the purpose of this exercise is to show how the difference between two Christoffel symbols that are asymmetric gives rise to torsion tensor.
 
  • #6
psimeson said:
@arkajad: Covariant derivative is not a hint in the problem. I am just trying to solve that way. I am following various kind of textbooks. So, any way would work for me.

Probably the simples way, for you, is to look at the transformation equations for the connection coefficients, and from that find out how the torsion will transform.

Check Eq. (3.6) in

http://preposterousuniverse.com/grnotes/grnotes-three.pdf

But do not read further than that!
 
Last edited by a moderator:
  • #7
Solved.
 

1. What is an antisymmetric connection?

An antisymmetric connection is a mathematical concept that describes the relationship between different points in a space or manifold. It is a type of connection that is defined by a torsion tensor, which measures the non-symmetric part of the connection.

2. How is the torsion tensor related to the antisymmetric connection?

The torsion tensor is the mathematical object that characterizes the antisymmetric connection. It is a type of tensor that measures the degree to which the connection is not symmetric, and it is defined in terms of the curvature tensor.

3. What are the applications of antisymmetric connection?

The antisymmetric connection has many applications in physics and mathematics. It is used in the study of general relativity, which is the theory of gravity, as well as in differential geometry and algebraic topology.

4. How does the torsion tensor affect the geometry of a space?

The torsion tensor plays a crucial role in defining the geometry of a space or manifold. It measures the deviation from symmetry in the connection, and this can have significant effects on the curvature and shape of the space.

5. Can the antisymmetric connection be visualized?

No, the antisymmetric connection is a mathematical concept that cannot be directly visualized. However, its effects on the geometry of a space can be visualized using various mathematical tools such as diagrams, graphs, and computer simulations.

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