Smooth maps with constant Jacobian determinant

Question

Can we say anything about smooth maps $F:\mathbb{R}^n \to \mathbb{R}^n$ whose Jacobian has constant non-zero determinant ? Are they dense in the space of smooth maps with respect to some metric ?

Answer 1

Note quite what you asked, but polynomial automorphisms of $\mathbb C^n$ must have constant Jacobian (indeed, their Jacobian is itself polynomial, and if it is non-constant then it must have zeroes somewhere in $\mathbb C^n$). There are many such automorphisms with real coefficients, which therefore are smooth automorphisms of $\mathbb R^n$ with constant Jacobian.

A famous conjecture states that conversely, any polynomial map with constant (non zero) Jacobian is an automorphism of $\mathbb C^n$.

I am skeptical about the density for any natural metric in that context.