Tag: map

Let $U subseteq mathbb R^2$ be an open, connected, bounded subset. Fix $0<sigma_1<sigma_2$, such that $sigma_1 sigma_2=1$.

Suppose that there exist a diffeomorphism $f:U to U$ such that the singular values of $df$ equal $sigma_1,sigma_2$ everywhere on $U$.

Question: Do there exist smooth isometries $
phi_1, phi_2:U to U$ such that $phi_1 circ f circ phi_2=f^{-1}$?

We must have $phi_i in operatorname{O}(2)$ (isometries are affine), but the point here is that I want them to map $U$ into $U$.

Note that $df^{-1}=(df)^{-1}$ has singular values $sigma_1,sigma_2$, the same as $df$.

Here are two examples where this phenomena happens:

1. Affine maps on ellipses:

Let $0<a<b$, $ab=1$, and let
$$
U=U_{a,b}=biggl{(x,y) ,biggm | , frac{x^2}{a^2} + frac{y^2}{b^2} < 1 biggr}.
$$

Take $f(x,y)=Apmatrix{x\y}$, where $$begin{align*} & A=A(theta)= begin{pmatrix} a& 0 \ 0 & b end{pmatrix} begin{pmatrix} costheta & -sintheta \ sintheta & cos theta end{pmatrix}begin{pmatrix} 1/a& 0 \ 0 & 1/b end{pmatrix}=
begin{pmatrix} costheta & -frac ab sintheta \ frac ba sintheta & cos theta end{pmatrix}
end{align*}.$$

Then $A(theta)^{-1}=A(-theta)=JA(theta) J$, where $J=begin{pmatrix} 1& 0 \ 0 & -1 end{pmatrix}$ is the reflection around the $y$ axis.

2. Non-affine maps on the disk:

Let $U=Dsetminus{0}$ where $D subseteq mathbb R^2$ is the unit disk.

$f_c: (r,theta)to (r,theta+clog r )$. Then we have $f_{c}^{-1}=f_{-c}=Jf_{c}J$.

Note that
$
(df_c)_{{ frac{partial}{partial r},frac{1}{r}frac{partial}{partial theta}}}=begin{pmatrix} 1 & 0 \ c & 1end{pmatrix},
$
so the singular values of $f_c$ are constants which depend on $c$.

Now, in general there are many local solutions to the PDE $sigma_i(df)=sigma_i$, so I don’t expect such a special relation between $f$ and its inverse in general. But I don’t have a counter-example yet.