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arXiv 2002.128842020-02-28math.DGmath.AGmath.APmath.ATmath.CV

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

Frederico Xavier

英文摘要：

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.

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