- #1

Mandelbroth

- 611

- 24

I found what might be the worst written book on Lie Groups. Ever. Until I find one I like better, I'm going to see if I can persevere through the sludge. I'll write out the theorem word for word and then explain what I can. Hopefully someone can decipher it.

Typically, I use the term "chart" to mean the inverse of what the author uses here. Also, in case it wasn't clear from the title, ##G## is a Lie group.

What does this mean, explicitly using the chart (preferably using ##\varphi_\alpha^{-1}## for his ##\varphi_\alpha##)?

Theorem: Let ##\varphi_\alpha: U_\alpha\to G_\alpha## be a chart on ##G## which sends ##0\in V## to ##\mathrm{e}\in G##. Then, omitting ##\varphi_\alpha##, we can write ##xy=x+y+o(r)##, where ##r## denotes the distance of ##(x,y)## from ##(\mathrm{e},\mathrm{e})## in ##G\times G## under a given metric.

Typically, I use the term "chart" to mean the inverse of what the author uses here. Also, in case it wasn't clear from the title, ##G## is a Lie group.

What does this mean, explicitly using the chart (preferably using ##\varphi_\alpha^{-1}## for his ##\varphi_\alpha##)?

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