Let $\mathbb{T}^2=\mathbb{S}^1 \times \mathbb{S}^1$ be the flat $2$-dimensional torus, and let $0<\sigma\_1 < \sigma\_2$ satisfy $\sigma\_1 \sigma\_2=1$.


Does there exist an area-preserving diffeomorphism $f:\mathbb{T}^2 \to \mathbb{T}^2$ whose singular values are constant $\sigma\_1 , \sigma\_2$?




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An immediate family of such diffeomorphisms which comes to mind are the affine (geodesic-preserving) maps which are induced by elements of $SL\_2(\mathbb{Z})$. However, this family does not cover the entire range of pairs $\{ (\sigma\_1,\frac{1}{\sigma\_1}) \, | \, \sigma\_1 \in (0,1) \}$, since it's countable. Furthermore the set of $\sigma\_1$ which are admissible in this affine family is discrete away from zero, which is its only accumulation point.
 
Are there any non-affine examples?




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**Edit:**


Robert Bryant has given an answer which shows that there is no non-affine $C^3$ example. I wonder what happens if we allow reduced regularity, say Lipschitz maps whose differential has a.e. the singular values $\sigma\_1 , \sigma\_2$.



A diffeomorphism of the torus with constant singular values
There are no non-affine examples with $f$ smooth (or even $C^3$). This follows from the fact that such an $f:\mathbb{T}^2\to\mathbb{T}^2$ would lift to a smooth (local) diffeomorphism $F:\mathbb{R}^2\to\mathbb{R}^2$ with constant singular values, and the argument I gave in my answer to [this question](https://mathoverflow.net/questions/351546/are-all-maps-mathbbr2-to-mathbbr2-with-fixed-singular-values-affine) implies that such a map $F$ would have to be affine.


