Journal of Mathematical Physics, Analysis, Geometry
2020, vol. 16, No 3, pp. 208-220   https://doi.org/10.15407/mag16.03.208     ( to contents , go back )
https://doi.org/10.15407/mag16.03.208

On Isometric Immersions of the Lobachevsky Plane into 4-Dimensional Euclidean Space with Flat Normal Connection

Yuriy Aminov

B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 47 Nauky Ave., Kharkiv, 61103, Ukraine
E-mail: aminov@ilt.kharkov.ua

Received April 30, 2020.

Abstract

According to Hilbert's theorem, the Lobachevsky plane $L^2$ does not admit a regular isometric immersion into $E^3$. The question on the existence of isometric immersion of $L^2$ into $E^4$ remains open. We consider isometric immersions into $E^4$ with flat normal connection and find a fundamental system of two partial differential equations of the second order for two functions. We prove the theorems on the non-existence of global and local isometric immersions for the case under consideration.

Mathematics Subject Classification 2000: 53C23, 53C45
Key words: isometric immersion, indicatrix, curvature, asymptotic line

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