It can be proved that all $\mathbb{R^2}\rightarrow\mathbb{R^2}$ mappings with constant singular values are affine. In three dimensions, however, there are non-trivial examples, like

$$ \begin{align} x'&=\lambda_2 x -\frac{1}{\lambda_2}\sqrt{\frac{\lambda_2^2-\lambda_1^2}{\lambda_3^2-\lambda_2^2}}\int_0^{z}\sin f(\xi)\,\mathrm{d}\xi \\ y'&=\lambda_2 y +\frac{1}{\lambda_2}\sqrt{\frac{\lambda_2^2-\lambda_1^2}{\lambda_3^2-\lambda_2^2}}\int_0^{z}\cos f(\xi)\,\mathrm{d}\xi \\ z'&=\frac{\lambda_1\lambda_3}{\lambda_2}z \end{align} $$

for an arbitrary differentiable $f$ and $\lambda_3>\lambda_2>\lambda_1$, whose differential has constant singular values $\lambda_3\,,\lambda_2\,,\lambda_1$. So I wonder if one can classify or say something generic about such $\mathbb{R^3}\rightarrow\mathbb{R^3}$ maps, as is the the case in two dimensions.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.