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.


1. Natanson I. P. Teoriya funkciy veschestvennoy peremennoy [Theory functions of real variable]. Moscow, Nauka, 1974. 480 p. (in Russian).
2. Lebesgue H. Sur le probleme de Dirichlet. Rend. Circ. Palermo, 1907, vol. 27, pp. 371–402.
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, pp. 53–104.
4. Vodop’yanov S. K. Monotone functions and quasiconformal mappings on Carnot groups. Siberian Math. J., 1996, vol. 37, no. 6, pp. 1113–1136. DOI: 10.1007/BF02106736.
5. Miklyukov V. M. Vvedenie v negladkiy analiz [Introduction in nonsmooth analysis]. Volgograd, Volgograd Univ. Press, 2008. 424 p. (in Russian).
6. Miklyukov V. M. Some conditions for the existence of the total differential. Siberian Math. J., 2010, vol. 51, no. 4, pp. 639–647. DOI: 10.1007/s11202-010-0065-9.
7. Salimov R. R. ACL and differentiability of a generalization of quasi-conformal maps. Izv. Math., 2008, vol. 72, no. 5, pp. 977–984. DOI: 10.1070/IM2008v072n05ABEH002425.
8. Prokhorova M. F. Problems of homeomorphism arising in the theory of grid generation. Proc. Steklov Inst. Math. (Suppl.), 2008, vol. 261, suppl. 1, pp. S165–S182. DOI: 10.1134/S0081543808050155.
9. Boluchevskaya A. V. On the Quasiisometric Mapping Preserving Simplex Orientation. Izv. Saratov Univ. (N.S.) Ser. Math. Mech. Inform., 2013, vol. 13, iss. 2, pp. 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), pp. 56–60 (in Russian). DOI: 10.15688/jvolsu1.2014.3.6.
11. Saks S. Teoriya integrala [Integral theory]. Moscow, Izd-vo. inostr. lit., 1949. 495 p. (in Russian).

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