Cite this article as:

Klyachin V. A., Chеbanеnko N. A. On the Geometric Structure of the Continuos Mappings Preserving the Orientation of Simplexes. Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics, 2017, vol. 17, iss. 3, pp. 294-303. DOI: https://doi.org/10.18500/1816-9791-2017-17-3-294-303

# On the Geometric Structure of the Continuos Mappings Preserving the Orientation of Simplexes

It is easy to show that if a continuous open map preserves the orientation of allsimplexes, the nit is affine. The class of continuous open maps f : D ⊂ R m → R n that preserve the orientation of simplexes from a given subset of a set of simplexes with vertices in the domain D ⊂ 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 possess ingthisproperty, one canobtainre sultsonits geometric structure. Inparticular, thepaper provesthat 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 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.

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