Transition Functions and Lie Groups

In summary, on Riemannian manifolds, transition functions are used to glue together coordinate charts, which are represented by Jacobian matrices. In the context of Lie groups and Fiber bundles, the process is the same, but with the added structure of these groups and bundles aiding in the gluing process. It is important to note that a Jacobian matrix is not a coordinate transformation, but rather built from the first derivatives of a coordinate transformation.
  • #1
knowwhatyoudontknow
30
5
I understand that on Riemannian manifolds, the transition functions that glue charts together are coordinate transformations (Jacobian matrices).

However, I am not quite sure how transition functions work in the context of Lie groups and Fiber bundles. Do we consider the manifolds to be flat and glue the charts together with elements of the Lie group whose actions represent coordinate translations?

On the other hand, if the manifold is flat, why do we need to consider more than a single chart?
 
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  • #2
knowwhatyoudontknow said:
I understand that on Riemannian manifolds, the transition functions that glue charts together are coordinate transformations (Jacobian matrices).
Just to be clear, a Jacobian matrix is not a coordinate transformation. It is built from the first derivatives of a coordinate transformation.

knowwhatyoudontknow said:
However, I am not quite sure how transition functions work in the context of Lie groups and Fiber bundles.
The same as any other manifold. You need to introduce local coordinate charts and glue them together. The particular thing with those is that they have some additional structure that may help you do that.
 
  • #3
Yes, I realize my error regarding the Jacobian matrices. I confused coordinate transformations with how functions change under a change of coordinates. This makes better sense now. Thanks.
 

FAQ: Transition Functions and Lie Groups

What is a transition function in the context of Lie groups?

A transition function is a mathematical function used to describe the smooth change between different local coordinate systems on a Lie group. It maps points from one coordinate system to another, allowing for a smooth transition between the two systems.

How are transition functions related to Lie groups?

Transition functions are essential in the study of Lie groups because they help define the local structure of the group. They allow us to smoothly move between different coordinate systems on a Lie group, which is crucial in understanding the group's properties and transformations.

Can transition functions be used to define differentiable manifolds?

Yes, transition functions are used in the definition of differentiable manifolds. In particular, they are used to define the smooth transition between different local charts on the manifold, which is necessary for the manifold to be differentiable.

How do transition functions relate to group actions?

Transition functions are closely related to group actions on a Lie group. In fact, they can be thought of as a special type of group action, where the group is acting on the coordinate systems of the group itself. This relationship is important in understanding the symmetries and transformations of a Lie group.

Are there any real-world applications of transition functions and Lie groups?

Yes, there are many real-world applications of transition functions and Lie groups. They are used in physics to describe symmetries and transformations of physical systems, in computer graphics to model smooth transformations, and in robotics for motion planning and control. They also have applications in other fields such as economics, engineering, and computer science.

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