[math] \newcommand{\mathds}{\mathbb}[/math]

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Work out the Weingarten formula for the classical spheres

[[math]] S^{N-1}_\mathbb R\quad,\quad S^{N-1}_\mathbb C [[/math]]

in general, then at small [math]N\in\mathbb N[/math], and at big [math]N\in\mathbb N[/math], and find some applications of this.