Euclidean Measure on Unit Sphere in $\Bbb{C}^n$

Question

What exactly is the Euclidean measure on the unit sphere in $\Bbb{C}^n$? How is it defined? What does it 'look' like?

Answer 1

I am not familiar with the term "Euclidean measure" but I would assume that this refers to the volume measure induced by the Euclidean metric. In other words, the Euclidean metric on $\mathbb{C}^n$ restricts to a Riemannian metric on the unit sphere $S^{2n-1}\subset\mathbb{C}^n$. This Riemannian metric then induces a measure in the usual way (via the Riemannian volume form in this case, since $S^{2n-1}$ is orientable).

As for how to think about this, it's just the usual "surface area" measure on a sphere. For instance, for $n=1$, this is the usual measure on a circle given by arc length. Another way to define it is that, up to a constant factor, the measure is defined as follows. Given $A\subseteq S^{2n-1}$, let $B=\{tx:x\in A,t\in[0,1]\}$. Then the measure of $A$ is (a certain constant factor times) the ordinary Lebesgue measure of $B$ as a subset of $\mathbb{C}^n$. In other words, the measure of a subset $A$ of the sphere is proportional to the Lebesgue measure of the "sector" $B$ of the unit ball you get by connecting $A$ to the origin radially.