Geodesic
On the geometric structure of the continuos mappings preserving the orientation of simplexes
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 17 (2017) no. 3, pp. 294-303 
V. A. Klyachin; N. A. Chеbanеnko. On the geometric structure of the continuos mappings preserving the orientation of simplexes. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 17 (2017) no. 3, pp. 294-303. http://geodesic.mathdoc.fr/item/ISU_2017_17_3_a5/
@article{ISU_2017_17_3_a5,
     author = {V. A. Klyachin and N. A. Ch{\cyre}ban{\cyre}nko},
     title = {On the geometric structure of the continuos mappings preserving the orientation of simplexes},
     journal = {Izvestiya of Saratov University. Mathematics. Mechanics. Informatics},
     pages = {294--303},
     year = {2017},
     volume = {17},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ISU_2017_17_3_a5/}
}
TY  - JOUR
AU  - V. A. Klyachin
AU  - N. A. Chеbanеnko
TI  - On the geometric structure of the continuos mappings preserving the orientation of simplexes
JO  - Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
PY  - 2017
SP  - 294
EP  - 303
VL  - 17
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/ISU_2017_17_3_a5/
LA  - ru
ID  - ISU_2017_17_3_a5
ER  - 
%0 Journal Article
%A V. A. Klyachin
%A N. A. Chеbanеnko
%T On the geometric structure of the continuos mappings preserving the orientation of simplexes
%J Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
%D 2017
%P 294-303
%V 17
%N 3
%U http://geodesic.mathdoc.fr/item/ISU_2017_17_3_a5/
%G ru
%F ISU_2017_17_3_a5

Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Natanson I. P., Theory functions of real variable, Nauka, M., 1974, 480 pp. (in Russian)

[2] Lebesgue H., “Sur le probleme de Dirichlet”, Rend. Circ. Palermo, 27 (1907), 371–402 | DOI

[3] Mostow G. D., “Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms”, Publ. Math. de l'lnstitute des Hautes Etudes Scientifiques, 1968, no. 34, 53–104 | DOI | MR | Zbl

[4] Vodop'yanov S. K., “Monotone functions and quasiconformal mappings on Carnot groups”, Siberian Math. J., 37:6 (1996), 1113–1136 | DOI | MR | Zbl

[5] Miklyukov V. M., Introduction in nonsmooth analysis, Volgograd Univ. Press, Volgograd, 2008, 424 pp. (in Russian)

[6] Miklyukov V. M., “Some conditions for the existence of the total differential”, Siberian Math. J., 51:4 (2010), 639–647 | DOI | MR | Zbl

[7] Salimov R. R., “ACL and differentiability of a generalization of quasi-conformal maps”, Izv. Math., 72:5 (2008), 977–984 | DOI | DOI | MR | Zbl

[8] Prokhorova M. F., “Problems of homeomorphism arising in the theory of grid generation”, Proc. Steklov Inst. Math. (Suppl.), 261, suppl. 1 (2008), S165–S182 | MR | Zbl

[9] Boluchevskaya A. V., “On the Quasiisometric Mapping Preserving Simplex Orientation”, Izv. Saratov Univ. (N. S.) Ser. Math. Mech. Inform., 13:1(2) (2013), 20–23 (in Russian)

[10] Klyachin V. A., Chebanenko N. A., “About linear preimages of continuous maps, that preserve orientation of triangles”, Science Journal of VolSU. Mathematics. Physics, 2014, no. 3 (22), 56–60 (in Russian)

[11] Saks S., Integral theory, Izd-vo. inostr. lit., M., 1949, 495 pp. (in Russian)