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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.

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.

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