Let $f: B \rightarrow \mathbb R^n$ be a diffrential map s.t $Df$ is non singular on $B$, where $B$ is an open ball in $\mathbb R^n$. What are some necessary and sufficient conditions by which $f$ is one-to-one on $B$? Also, if I replace $B$ with an arbitrary open set, what are the conditions then?
Thanks a lot, Gil.
For $n=1$ this is automatic essentially by the fundamental theorem of calculus. However, for $n\geq 2$ there is no local condition that would ensure injectivity. Consider for example the restriction of the exponential map $z\mapsto e^z$ to a narrow open strip around the imaginary axis. The strip gets wrapped around a neighborhood of the unit circle and there is no hope of injectivity unless a global condition is added.