Maps That Must Be Affine or Conjugate Affine: A Problem of Vladimir Arnold

Abstract

A k-flat in a vector space is a k-dimensional affine subspace. Our basic result is that an injection $$T:{{\mathbb {C}}}^n\rightarrow {{\mathbb {C}}}^n$$ that for some $$k\in \{1,2,\ldots ,n-1\}$$ , T maps all k-flats to flats of $${{\mathbb {C}}}^n$$ and is either continuous at a point or Lebesgue measurable, is either an affine map or a conjugate-affine map. An analogous result is proven for injections of the complex projective spaces. In the case of continuity at a point, this is generalized in several directions, the main one being that the complex numbers can be replaced by a finite-dimensional division algebra over an Archimedean ordered field. We also prove injective versions of the Fundamental Theorems of affine and projective geometry and give a counter-example to the surjective version of the latter. This extends work of A. G. Gorinov on a problem of V. I. Arnold.