I Transition Functions and Lie Groups

[knowwhatyoudontknow]

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?

 

[Orodruin]

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.

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.

 

[knowwhatyoudontknow]

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.