Every continuous probability distribution $p$ with support $\mathbb{R}^n$ can be represented by a standard $n$-variate normal distribution transformed by some bijective, differentiable function $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$, that is, $p(x)=g(f^{-1}(x))|\det J_{f^{-1}}(x)|$ where $g$ is the density of a standard $n$-variate Gaussian and $J_{f^{-1}}(x)$ is the Jacobian of $f^{-1}$ evaluated at $x$.
Is the same still true if we restrict $f$ to be volume preserving? That is, can every continuous distribution with support $\mathbb{R}^n$ be obtained by transforming a standard Gaussian by a bijective differentiable function $f$ such that $|\det J_{f^{-1}}|=1$?
Thanks!
I realized that the answer is trivially no as in the simple 1 dimensional case, the only differentiable, invertible, volume-preserving transformations are of the form $f(x) = x + c$ or $f(x)=-x + c$.