  It is shown that in dimension at least three a local diffeomorphism of
Euclidean n-space into itself is injective provided that the pull-back of every
plane is a Riemannian submanifold which is conformal to a plane. Using a
similar technique one recovers the result that a polynomial local
biholomorphism of complex $n$-space into itself is invertible if and only if
the pull-back of every complex line is a connected rational curve. These
results are special cases of our main theorem, whose proof uses geometry,
complex analysis, elliptic partial differential equations, and topology.
