Regard $S^n$ and $S^1 \times \mathbb{R}^{n-1}$ as codimension-$1$ submanifolds of $\mathbb{R}^{n+1}$, equipped with the volume forms they inherit from $\mathbb{R}^{n+1}$. Letting $X = S^n \setminus 0 \times S^{n-2}$, define $f\colon X \rightarrow S^1 \times \mathbb{R}^{n-1}$ via the formula $$f(x_1,\ldots,x_{n+1}) = \left(\frac{x_1}{\sqrt{x_1^1+x_2^2}},\frac{x_2}{\sqrt{x_1^1+x_2^2}},x_3,\ldots,x_{n+1}\right).$$ It goes back to Archimedes that $f$ is a volume-preserving map onto its image. I can do the calculation with the formulas and verify this, but it seems mysterious to me. Does anyone know a more geometric or conceptual explanation? There are vague descriptions all over the web (for instance, in the "geometric proof" of this wikipedia article), but I have trouble turning these into precise mathematics.
3
-
$\begingroup$ Does this help? mathoverflow.net/questions/113780/… $\endgroup$– Steven LandsburgCommented Oct 15, 2022 at 23:47
-
$\begingroup$ @StevenLandsburg: David Speyer's answer is basically what I was looking for. Thanks! $\endgroup$– XiyanCommented Oct 16, 2022 at 0:20
-
$\begingroup$ These days, it's a combination of simple geometry and simple trigonometry. However, Van der Waerden presented the two original Archimedes proofs, beautiful! One based on mechanics, followed by a purely geometric argument. Of course, this is true only for $S^2$, not for other dimensions. $\endgroup$– Wlod AACommented Oct 16, 2022 at 1:04
Add a comment
|