On Preservation of the Sign of Gaussian Curvature Under Geodetic Mappings
On Preservation of the Sign of Gaussian Curvature Under Geodet
DOI:
https://doi.org/10.56143/ujmcs.v1i1.14/Keywords:
smooth manifolds; affine connection; geodesic curve; geodesic mapping; non-trivial geodesic mapping; Gaussian curvature.Abstract
Geodesic mappings have important applications in Riemannian geometry, in the theory of geodesy and cartography, modeling, physics and mechanics. In this paper, the question of preserving the sign of Gaussian curvature under geodesic mappings is investigated. It is proved that if surfaces of revolution have constant Gaussian curvature, then under a nontrivial geodesic mapping the sign of Gaussian curvature is preserved.
References
[1] Синюков Н. Геодезические отображения римановых пространств. М.: Наука, 1979. – 245с.
[2] Chud´a H., Mikeˇs J., Sochor M. Rotary diffeomorphism onto manifolds with affine connections. Geometry, Integrability and Quantization 11, 2017, 130–137.
[3] Hinterleitner I. On global geodesic mappings of ellipsoids. AIP. Conf. Proc. 1460, 2012, 180–184.
[4] Синюкова Е.Н., Чепок О.Л. О геодезических отображениях в целом римановых про странств, удовлетворяющих некоторым условиям дифференциально-алгебраического харак тера // Известия ПГПУ им. В.Г. Белинского. 2011. № 26. С. 214–221.
[5] Josef Mikeˇs, Elena Stepanova, Alena Vanˇzurov´a, Differential Geometry of Special Mappings. et al. First Edition Palack´y University, Olomouc, 2015.
[6] Abdullaaziz Artıkbayev, Abdullah Kurudirek, H¨useyin Ak¸ca. Occurrence of Galilean Geometry. Applied and Computational Mathematics, Vol. 2, No. 5, 2013, pp. 115-117. doi: 10.11648/j.acm.20130205.11
[7] Ismoilov Sh., Sultanov B. Invariant Geometric Characteristics Under the Dual Mapping of an Isotropic Space, Asia Pacific Journal of Mathematics, 10(1), 20, 2023.
[8] Sh. Ismoilov, B. Sultanov, B. Mamadaliyev, A. Kurudirek. Translation surfaces with non-zero constant total curvature in multidimensional isotropic space, J. Appl. Math. Informatics Vol. 43(2025), No. 1, pp. 191 – 203
[9] Sharipov A., Keunimjaev M. Existence and uniqueness of polyhedra with given values of the conditional curvature, International Electronic Journal of Geometry, 2023, 16(1), pp.160-170.
[10] Topvoldiyev F. Conditional external curvatures of irregular cones. Bull. Inst. Math., 2023, Vol.6, No 3, pp. 34-41
[11] Topvoldiyev F., Sharipov A. On Defects of Polyhedra Isometric on Section at Vertics, AIP Conference Proceedings, 2024, 3004(1), 030011.
[12] John Oprea. Differential Geometry and its applications. The Mathematical Association of America, 2007.
[13] Jean Gallier, Jocelyn Quaintance. Differential Geometry and Lie Groups. Springer Nature Switzerland AG 2020.
[14] Lenka Ryparova. Geodesics and their mappings. Ph.D. thesis. 2020.
