We introduce the notion of Leiko network as a generalization of the geodetic network on the surfaces of nonzero Gaussian curvature in the Euclidian space E 3 and study its characteristics. The conditions of preservation of the Leiko network under infinitesimal deformations of the surfaces are also obtained.
Similar content being viewed by others
References
L. L. Bezkorovaina, Areal Infinitely Small Deformations and Steady States of Elastic Shells [in Ukrainian], AstroPrynt, Odesa (1999).
V. F. Kagan, Foundations of the Theory of Surfaces in Tensor Representation. Part 1 [in Russian], OGIZ, Moscow (1947).
V. F. Kagan, Foundations of the Theory of Surfaces in Tensor Representation. Part 2 [in Russian], OGIZ, Moscow (1948).
S. G. Leiko, “Theorem on existence of the extremes of rotation on surfaces in E 3 and rotational diffeomorphisms,” in: All-Union School. Optimal Control. Geometry and Analysis [in Russian], Kemerovo (1988), p. 52.
S. G. Leiko, “Variational problems for functionals of rotation and spin mappings of pseudo-Riemannian spaces,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 10, 9–17 (1990).
S. G. Leiko, “Rotational diffeomorphisms on the surfaces in Euclidean space,” Mat. Zametki, 47, No. 3, 52–57 (1990).
S. G. Leiko, “Extremes of functionals of rotation of the curves of a pseudo-Riemannian space and the paths of spin-particles in gravitational fields,” Dokl. Ros. Akad. Nauk, 325, No. 4, 659–664 (1992).
S. G. Leiko, “Isoperimetric extremes of rotation on surfaces in the Euclidean space E 3 ,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 6, 25–32 (1996).
I. V. Potapenko, “On the reconstruction of the variation of the metric tensor of a surface on the basis of a given variation of Christoffel symbols of the second kind under infinitesimal deformations of surfaces in the Euclidean space E 3 ,” Ukr. Mat. Zh., 63, No. 4, 523–530 (2011); English translation: Ukr. Math. J., 63, No. 4, 609–616 (2011).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 6, pp. 820–828, June, 2015.
Rights and permissions
About this article
Cite this article
Potapenko, I.V. Leiko Network on the Surfaces in the Euclidean Space E 3 . Ukr Math J 67, 928–937 (2015). https://doi.org/10.1007/s11253-015-1123-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-015-1123-z