I would like to compute the following integral (with respect to Lebesgue measure): $$ \int_{x\in S(\mathbb{C}^2)} \langle x | y\rangle x $$
- Here $x\in \mathbb{C}^2$ lies on the unit hypershere of $\mathbb{C}^2$ which is alternatively the real 3-sphere $S^3$.
- $y \in \mathbb{C}^2$ as well.
- $\langle x|y\rangle = \bar{x_1}y_1 + \bar{x_2}y_2 $ is the inner product of $\mathbb{C}^2$
So the domain of this integral is the real 3-sphere and it is valued in $\mathbb{C}^2$.
How should I proceed? What parametrization of the 3-sphere should I use? The Hopf coordinates seem like a good option but are they even injective/differentiable? Will I be forced to split the sphere into several pieces so that the parametrization remains injective and differentiable for each piece (I am wanting to apply this theorem)?
I feel like the answer should be proportional to $y$. Ideally I would like to compute this integral for $\mathbb{C}^n$ instead of just $\mathbb{C}^2$.
EDIT: partially answering my question: The answer is indeed proportional to $y$ because for any $x$ in the domain you can find a unique $x'$ such that $$\langle x | y \rangle x + \langle x' | y \rangle x' = cy \quad (c\in \mathbb{C})$$ So the integral only adds up vectors lying on the $y$ subspace and is consequently proportional to $y$ itself. I suspect that the factor in front of $y$ is real and correspond to the area of the hypersphere $S(\mathbb{C^2})$.
Consider the function $f:\Bbb{C}^n\to\Bbb{C}^n$ \begin{align} f(y)&:=\int_{S(\Bbb{C}^n)}\langle x|y\rangle x\, d\sigma(x), \end{align} where $\sigma$ is the surface measure on $S(\Bbb{C}^n)\cong S^{2n-1}$. By your convention for the inner product, $f$ is linear so it is completely determined by its values on the basis vectors $e_a=(0,\dots, 1,\dots 0)\in\Bbb{C}^n$. Now, \begin{align} f(e_a)&=\int_{S(\Bbb{C}^n)}\overline{x_a}\cdot x\,\,d\sigma(x)= \left(\int_{S(\Bbb{C}^n)}\overline{x_a}x_1\,d\sigma(x),\dots, \int_{S(\Bbb{C}^n)}\overline{x_a}x_n\,d\sigma(x)\right) \end{align} Thus, we're reduced to calculating the integrals $\int_{S(\Bbb{C}^n)}\overline{x_a}x_b\,d\sigma(x)$ for all $a,b\in\{1,\dots, n\}$.
If $a\neq b$, then the integral is zero, because if we consider the function $(x_1,\dots, x_n)\mapsto(x_1,\dots, -x_a,\dots, x_n)$, then the change of variables theorem introduces a minus sign in the integrand while keeping the measure and the domain $S(\Bbb{C}^n)$ the same, hence the integral is $0$ (this is the higher dimensional analogue of the fact that the integral of an odd function over a symmetric interval vanishes).
Suppose now that $a=b$, so that we're considering $\int_{S(\Bbb{C}^n)}|x_a|^2\,d\sigma(x)$. Again, by symmetry, we have \begin{align} \int_{S(\Bbb{C}^n)}|x_1|^2\,d\sigma(x)=\dots =\int_{S(\Bbb{C}^n)}|x_n|^2\,d\sigma(x). \end{align} Thus, \begin{align} \int_{S(\Bbb{C}^n)}|x_a|^2\,d\sigma(x)&=\frac{1}{n}\int_{S(\Bbb{C}^n)}\sum_{j=1}^n|x_j|^2\,d\sigma(x)\\ &=\frac{A_{2n-1}}{n}\\ &=\frac{2\pi^n/\Gamma(n)}{n}\\ &=\frac{2\pi^n}{\Gamma(n+1)}\\ &=\frac{2\pi^n}{n!}, \end{align} where $A_{k-1}=\frac{2\pi^{k/2}}{\Gamma(k/2)}$ is the surface area of the unit sphere $S^{k-1}\subset\Bbb{R}^k$ (we have also used some basic properties of the Gamma function). In other words, putting it all together, we have shown that \begin{align} f(e_a)&=\frac{A_{2n-1}}{n}e_a=\frac{2\pi^n}{n!}e_a, \end{align} and thus \begin{align} f(y)&=\frac{A_{2n-1}}{n}y=\frac{2\pi^n}{n!}y. \end{align}