Let $\left(\mathcal{M}^2,g_\mathcal{M};X\right)$ and $\left(\mathcal{N}^2,g_{\mathcal{N}};Y\right)$ be two smooth two-dimensional, simply connected Riemannian manifolds (with or without boundary), equipped with non-nonvanishing Killing fields $X$ and $Y$, respectively.

Does there exist a mapping $f:\mathcal{M}^2\rightarrow \mathcal{N}^2$ with constant, distinct singular values such that $\mathrm{d}f:X\rightarrow Y$ ?

In the affirmative case, how would one construct explicitly such a mapping, given the singular values ?

(It is well known that in general there are no local obstructions for mapping with constant singular values, but assuming that there exists a Killing field that is mapped to another Killing may set obstructions.)

Let $\left(\mathcal{M}^2,g_\mathcal{M};X\right)$ and $\left(\mathcal{N}^2,g_{\mathcal{N}};Y\right)$ be two smooth two-dimensional, simply connected Riemannian manifolds (with or without boundary), equipped with non-nonvanishing Killing fields $X$ and $Y$, respectively.

Does there exist a mapping $f:\mathcal{M}^2\rightarrow \mathcal{N}^2$ with constant, distinct singular values such that $\mathrm{d}f\left(X\right)=Y$ ?

In the affirmative case, how would one construct explicitly such a mapping, given the singular values ?

(It is well known that in general there are no local obstructions for mapping with constant singular values, but assuming that there exists a Killing field that is mapped to another Killing may set obstructions.)