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

Conformal geometry, Euler numbers, and global invertibility in higher dimensions