# Manifolds / Lie Groups - confusing notation

1. Mar 16, 2008

### GSpeight

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

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 $F_{\star}$ is the map from M to $d_{x}F$. 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, $(L_{g})_{\star}X_{h}=X_{gh}=X_{L_{g}(h)}$ where $L_{g}$ is the left multiplication map by g ,which I'm finding difficult to understand with my current definition of $F_{\star}$.

Thanks for any replies.

Last edited: Mar 17, 2008