Example of Bianchi Transformation of Kuen’s Surface
УДК 514.7
Abstract
The work is devoted to the study of the Bianchi transformation for surfaces of constant negative Gaussian curvature. The surfaces of rotation of constant negative Gaussian curvature are the Minding top, the Minding coil, and the pseudosphere (Beltrami surface). Surfaces of constant negative Gaussian curvature also include Kuen’s surface and the Dini’s surface. Studying the surfaces of constant negative Gaussian curvature (pseudospherical surfaces) is of great importance for the interpretation of Lobachevsky planimetry. Geometric characteristics of pseudospherical surfaces are found to be related to the theory of networks, the theory of solitons, nonlinear differential equations, and sin-Gordon equations. The sin-Gordon equation plays an important role in modern physics. Bianchi transformations make it possible to obtain new pseudospherical surfaces from a given pseudospherical surface. The Bianchi transformation for the Kuen’s surface is constructed using a mathematical software package.
Downloads
Metrics
References
Popov A.G. Psevdospherical surfaces and some problems of matemacal physics // Fundamentalnaya i prikladnaya mathematiks. 2005. Vol. 11. № 1.
Поздняк Э.Г., Попов А.Г. Геометрия Лобачевского и уравнения математической физики // Докл. РАН. 1993. T. 332. № 4.
Аминов Ю.А. Преобразование Бианки для области многомерного пространства Лобачевского // Украинский геометрический сборник. 1978. Т.21.
Tenenblat K. Transformations of manifolds and applications to differential equations. Pseudospherical surfaces and some problems of matematical physics. London, 1998.
Горькавый В.А., Невмержицкая Е.Н. Аналог преобразования Бианки для двумерных поверхностей в пространстве S3XR1 // Матем. заметки. 2011. Т.89. № 6.
Масальцев Л.А. Бикасательное преобразование Бианки подмногообразия постоянной отрицательной кривизны Hn евклидова пространства R2n // Изв. вузов. Матем. 2005. T. 7.
Каган В.Ф. Основы теории поверхностей в тензорном изложении. М., 1947. Т. II.
Норден А.П. Об основаниях геометрии. М., 1956.
Чешкова М.А. Геодезические поверхности вращения постоянной гауссовой кривизны // Сибирский журнал чистой и прикладной математики. 2018. T. 18. № 3.
Чешкова М.А. Преобразование Бианки n -поверхностей в E2n-1 // Изв. вузов. Матем. 1997. T. 9.
Шуликовский В.И. Классическая дифференциальная геометрия в тензорном изложении. М., 1963.
Кривошапко С.Н., Иванов В.Н., Халаби С.М. Аналитические поверхности. М., 2006.
Izvestiya of Altai State University is a golden publisher, as we allow self-archiving, but most importantly we are fully transparent about your rights.
Authors may present and discuss their findings ahead of publication: at biological or scientific conferences, on preprint servers, in public databases, and in blogs, wikis, tweets, and other informal communication channels.
Izvestiya of Altai State University allows authors to deposit manuscripts (currently under review or those for intended submission to Izvestiya of Altai State University) in non-commercial, pre-print servers such as ArXiv.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License (CC BY 4.0) that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
