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.
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Acknowledgements
We wish to thank the referee for suggesting that the proof of the first part of Theorem 2 should extend to arbitrary topological vector spaces. As a result, we have been able to state and prove our Main Theorems in greater generality.
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George F. McNulty was supported by NSF Grant 1500216.
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Gorinov, A., Howard, R., Johnson, V. et al. Maps That Must Be Affine or Conjugate Affine: A Problem of Vladimir Arnold. Arnold Math J. 6, 213–229 (2020). https://doi.org/10.1007/s40598-020-00147-7
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DOI: https://doi.org/10.1007/s40598-020-00147-7