Does anybody know how to write the charts for the special 2x2 unitary matrices SU(2) as a manifold. i.e. those satisfying GG*=I where G* is the conjugate transpose and I is the identity matrix. This problem has been bugging me for ages, would be very very gratefull for any help....
I can use hyperspherical coordinates to express a,b,c,d in 3 variables as follows:
b=cosG sinH sinF
c=sinG sinH sinF
This means that the manifold has dimension 3. I have read that I will need atleast 2 charts to cover the whole manifold, however wont the chart mapping (F,G,H) to (Z=a+ib , W=c+id)
(-W* , Z* )
cover the whole manifold? If not can you please show me the two charts needed? Thankyou very much for your help, this problem has been bugging me for weeks now..