Evaluating a surface integral Denote by $m$ the surface measure on the unit sphere $B$ in $\mathbb{C}^2$. Consider an unitary $2\times 2$ matrix $U$ and suppose we have a function $f\colon B\to \mathbb{C}$ which is $m$-integrable. Is it true that
$\int_B f(x)\,\mbox{d}x = \int_B f(Ux)\,\mbox{d}x?$

Unitary matrices $U$ have the property that $<x,y>=<Ux,Uy>$, meaning they preserve the inner-product , i.e., the metric . Using a Real version of the Jacobian 
(since |Det(U)|=1) , may also give you an argument to this effect.
