I am interested in obtaining injectivity of a $C^1$ map from the nonvanishing minors of its Jacobian matrix. Here is a brief history of the topic.

In 1953, Samuelson asked the following:

If the upper left-hand principal minors of the Jacobian matrix of a map $F: \mathbb{R}^n\rightarrow \mathbb{R}^n$ do not vanish, is it true that $F$ must be injective?

In 1965, Gale and Nikaido gave a counterexample in $\mathbb{R}^2$. In their paper the following is proved

Gale-Nikaido theorem:If all the principal minors of the Jacobian matrix of $F: \mathbb{R}^n\rightarrow \mathbb{R}^n$ arepositive, then $F$ is injective.

Since then, some effort has been made to weaken the assumption in Gale-Nikaido theorem since the assumption seems to be too restrictive in application. A comprehensive dicussion can be found in T. Parthasarathy, *On Global Univalence Theorems*, Lecture Notes in Mathematics, Vol. 977, 1983. In the case of polynomial map, this is related to the real version of Jacobian conjecture.

A possible generalization I'm interested in is the following, which seems to be open.

Question:If all the principal minors of the Jacobian matrix of $F: \mathbb{R}^n\rightarrow \mathbb{R}^n$do not vanish, is $F$ necessarily injective?

In Gale and Nikaido's paper, the case of $\mathbb{R}^2$ was answered in affirmative, the case of $\mathbb{R}^3$ was claimed in affirmative (yet no complete proof seems to be known).

My motivation comes from trying to make a change of variables to globally rectify a curved coordinate system so that Plancherel theorem can be applied. Any information would be appreciated : )

Zorich Theorem(en.wikipedia.org/wiki/Zorich's_theorem): If $f: {\mathbb R}^n\to {\mathbb R}^n$ is a locally-injective quasiregular map, then, for $n\ge 3$, the map $f$ is a homeomorphism. A smooth map $f$ is $K$-quasiregular if $||Df(x)||^n\le K |J_f(x)|$ for all $x\in {\mathbb R}^n$. The assumptions are somewhat different from the ones you are asking, but the conclusion is the same. – Misha May 7 '12 at 4:34