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 Izv. Saratov Univ. Math. Mech. Inform., 2017, Volume 17, Issue 3, Pages 294–303 (Mi isu725)

Scientific Part
Mathematics

On the geometric structure of the continuos mappings preserving the orientation of simplexes

V. A. Klyachin, N. A. Chåbanånko

Abstract: It is easy to show that if a continuous open map preserves the orientation of all simplexes, then it is affine. The class of continuous open maps $f: D \subset \mathbb R^m \to \mathbb R^n$ that preserve the orientation of simplexes from a given subset of a set of simplexes with vertices in the domain $D \subset \mathbb R^m$ is considered. In this paper, questions of the geometric structure of linear inverse images of such mappings are studied. This research is based on the key property proved in the article: if a map preserves the orientation of simplexes from some subset $B$ of the set of all simplexes with vertices in the domain $D$, then the inverse image of the hyperplane under such a mapping can not contain the vertices of a simplex from $B$. Based on the analysis of the structure of a set possessing this property, one can obtain results on its geometric structure. In particular, the paper proves that if a continuous open map preserves the orientation of a sufficiently wide class of simplexes, then it is affine. For some special classes of triangles in $\mathbb R^2$ with a given condition on its maximal angle it is shown that the inverse image of a line is locally a graph (in some case a Lipschitzian) of a function in a suitable Cartesian coordinate system.

Key words: simplex, orientation of simplex,continuous mapping, monotone function.

 Funding Agency Grant Number Russian Foundation for Basic Research 15-41-02517-ð_ïîâîëæüå_à This work was supported by the Russian Foundation for Basic Research (project no. 15-41-02517).

DOI: https://doi.org/10.18500/1816-9791-2017-17-3-294-303

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UDC: 514.17

Citation: V. A. Klyachin, N. A. Chåbanånko, “On the geometric structure of the continuos mappings preserving the orientation of simplexes”, Izv. Saratov Univ. Math. Mech. Inform., 17:3 (2017), 294–303

Citation in format AMSBIB
\Bibitem{KlyChå17} \by V.~A.~Klyachin, N.~A.~Chåbanånko \paper On the geometric structure of the continuos mappings preserving the orientation of simplexes \jour Izv. Saratov Univ. Math. Mech. Inform. \yr 2017 \vol 17 \issue 3 \pages 294--303 \mathnet{http://mi.mathnet.ru/isu725} \crossref{https://doi.org/10.18500/1816-9791-2017-17-3-294-303} \elib{https://elibrary.ru/item.asp?id=29897302} 

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This publication is cited in the following articles:
1. V. A. Klyachin, N. A. Chebanenko, “O geometricheskikh svoistvakh nepreryvnykh otobrazhenii, sokhranyayuschikh orientatsiyu simpleksov”, Sib. elektron. matem. izv., 18:2 (2021), 985–996
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