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Let $\mathbb{T}^2$ be the flat torus with Haar measure $\mu_{\mathbb{T}^2}$, and $\mathbb{S}^2$ the sphere with the spherical (rotationally-invariant) measure $\mu_{\mathbb{S}^2}$.

Is is possible to construct a smooth (or even just Lipchitz-continuous) surjection $\Theta:\mathbb{T}^2\to \mathbb{S}^2$ such that $\mu_{\mathbb{T}^2}\circ \Theta^{-1}=\mu_{\mathbb{S}^2}$? i.e. for any measurable set $A\subseteq \mathbb{S}^2$, $\mu_{\mathbb{T}^2}(\Theta^{-1}(A))=\mu_{\mathbb{S}^2}(A)$.

It is indeed possible to construct smooth surjections from the torus to the sphere (e.g. this question). It seems to me that those surjections cannot possibly preserve the measures.

For example, a natural map is to send the torus to a cylinder, and then map the cylinder into a sphere, but the equal-area projection from a cylinder into the sphere is not Lipchitz continuous at the poles.

I would appreciate some references on the matter, thanks!

Saúl Pilatowsky-Cameo
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  • Haar measure is only defined on Lie or topological groups, no? – Ted Shifrin Sep 05 '22 at 22:36
  • @TedShifrin Haar measure can be defined on any homogeneous space. In this case $SO(3)$ acts on the sphere and $SO(2)\times SO(2)$ acts on the torus, so they naturally inherit a Haar measure: [For constructions see https://math.stackexchange.com/questions/3720530/what-is-the-haar-measure-on-the-unit-sphere and https://math.stackexchange.com/questions/4343890/constructing-the-haar-measure-of-the-n-dimensional-torus ]. – Saúl Pilatowsky-Cameo Sep 06 '22 at 00:22
  • I know perfectly well how to construct invariant metrics, forms, and measures on homogeneous spaces. I’m complaining about the terminology. Citing MSE questions doesn’t help convince me. P.S. The torus is a group. – Ted Shifrin Sep 06 '22 at 00:52
  • @TedShifrin I just needed a name to refer to the natural measures on the sphere and the flat torus. Maybe Haar measure is not the best terminology. As you say, the torus is a [compact] group so Haar measure makes sense. As for the sphere, maybe spherical measure is a better term? I'll edit it in. – Saúl Pilatowsky-Cameo Sep 06 '22 at 01:05
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    “Invariant” is the usual term, the “usual” homogeneous structure being understood. – Ted Shifrin Sep 06 '22 at 01:08

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