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Manifolds / Lie Groups - confusing notation

  • Thread starter GSpeight
  • Start date
31
0
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.
 
Last edited:

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