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

V. A. Klyachin ; N. A. Chеbanеnko

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 17 (2017) no. 3, pp. 294-303

Résumé

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.