Re: Question on Lie Groups

igor.kh_at_gmail.com
Date: 01/31/05 

Date: 30 Jan 2005 16:45:37 -0800

Tony wrote:
> Hi everyone,
>
> Doing some problems on my own outside of class, and can't seem to get
the
> following :
>
> Let G be a connected Lie group, and let U in G be any open
neighborhood of
> the identity. Show that every element of G can be written as a
finite
> product of elements of U.
>
> I can't seem to figure this out.

How about this. Since G is connected, there is a continuous curve from
the identity to any element. Strictly speaking, this requires path
connectedness, but I think for manifolds they are equivalent. This
curve is homeomorphic to the unit interval [0,1] and hence is compact.
If U is a neighborhood of the identity, then cover this curve by left
(or right) translations of U by each group element lying on the curve.
Since the curve is compact, a finite subcover can be selected. Now
leapfrog between overlapping neighborhoods.

Igor

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