Is the orthogonal polar factor the unique retraction $\operatorname{GL}_n^+ \to \operatorname{SO}_n$?

Question

$\newcommand{\psym}{\text{Psym}_n}$ $\newcommand{\sym}{\text{sym}}$ $\newcommand{\Sym}{\operatorname{Sym}}$ $\newcommand{\Skew}{\operatorname{Skew}}$ $\renewcommand{\skew}{\operatorname{skew}}$ $\newcommand{\GLp}{\operatorname{GL}_n^+}$ $\newcommand{\SO}{\operatorname{SO}_n}$

This might be silly, but I wonder:

Let $F:\GLp \to \SO$ be a continuous retract. Is it true that $F$ must be the orthogonal polar factor, i.e. $F(A)=O$, where $A=OP,O \in \SO,P\in\psym$. Does anything changes if we assume $F$ is a deformation retract? Or if it is a smooth deformation retract?

Answer 1

Retractions are a very "soft" concept so there is a huge number of those. Take any continuous function $f:PSym_n\to SO_n$ and define $F_f(A):=O\cdot f(P)$ for $A=OP$. Then this is clearly a retraction and it even is a defomormation retraction, which is smooth if $f$ is smooth.