On geometrical properties of continuous mappings which preserve orientation of simplices

It is easy to show that if a continuous and open mapping preserves the orientation of all simplices, then it is affine. The article discusses the class of continuous, open mappings $ f: D \subset \mathbb{R}^3 \to \mathbb{R}^3$ that preserve the orientation of simplices from a given subset of the set of simplexes with vertices in the domain $ D \subset \mathbb{R}^3 $. In this paper, the questions of the geometric structure of linear transforms of such mappings are investigated. This study is based on a key property: if a map preserves the orientation of simplices from a certain subset $ B $ of the set of all simplices with vertices in $ D $, then the pre-image of a hyperplane cannot contain vertices of a simplex from $ B $. Based on the analysis of the structure of a set with such a property, it is possible to obtain results on its geometric structure. In particular, the article proved that if a continuous and open mapping preserves the orientation of a fairly wide class of simplices, then it is affine. For some special classes of triangles in $ \mathbb{R}^2 $ with a given condition on its maximum angle, the authors previously proved that the inverse image of a line is locally a graph of a function (in some case, Lipschitz) in a suitable Cartesian coordinate system.