Manifolds / Lie Groups - confusing notation

[GSpeight]

I'm reading over my Lie groups notes and in them, in the introductory section on manifolds, I've written that [itex]F_{\star}[/itex] is a commonly used notation for [itex]d_{x}F[/itex] and so the chain rule [itex]d_{x}{G \circ F}=d_{F(x)}G \circ d_{x}F[/itex] can be written [itex](G\circ F)_{\star}=G_{\star}\circ F_{\star}[/itex]

Is what I've written correct? To me this seems horribly confusing since it neglects to mention where you are taking the differential. Should it instead be that [itex]F_{\star}[/itex] is the map from M to [itex]d_{x}F[/itex]. On second thoughts this doesn't make total sense either...

He's gone on to make definitions like:

A vector field X on a Lie group G is called left-invariant if, for all g,h in G, [itex](L_{g})_{\star}X_{h}=X_{gh}=X_{L_{g}(h)}[/itex] where [itex]L_{g}[/itex] is the left multiplication map by g ,which I'm finding difficult to understand with my current definition of [itex]F_{\star}[/itex].

Thanks for any replies.

 

[mighty2000]

Thank you for sharing your thoughts and questions on Lie groups and manifolds. From what you have written, it seems like you have a good understanding of the notation and concepts involved. However, I would like to clarify a few things for you and provide some additional insights.

Firstly, you are correct in saying that F_{\star} is commonly used to denote the differential of the map F at a point x. This notation is often used in differential geometry and is a shorthand way of writing d_{x}F. So, when we write (G\circ F)_{\star}=G_{\star}\circ F_{\star}, we are essentially saying that the differential of the composition of two maps is equal to the composition of their differentials. This is known as the chain rule and is a fundamental concept in differential calculus.

However, I can understand your confusion when it comes to the differential of a map. It is important to specify where we are taking the differential, as it can vary depending on the context. In the case of manifolds, we are usually interested in the differential at a specific point, which is denoted by d_{x}F or F_{\star}. But in other contexts, we may be interested in the differential along a curve or a tangent vector, which would be denoted differently.

Moving on to the definition of left-invariant vector fields on a Lie group, I can see why you are finding it difficult to understand with your current definition of F_{\star}. Let me try to explain it in a different way. A left-invariant vector field X on a Lie group G is a vector field that remains unchanged under left translations. In other words, if we apply the left translation map L_{g} to the vector field X, it will still be the same vector field. This can be expressed mathematically as (L_{g})_{\star}X=X. Now, using the chain rule, we can write (L_{g})_{\star}X_{h}=X_{L_{g}(h)}. This means that the differential of X at the point h is equal to the value of X at the point L_{g}(h). This is what the definition is trying to convey.

I hope this helps you understand the concepts better. Don't hesitate to ask if you have any further questions. Keep up the good work with your studies on Lie groups and