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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

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.

Key words and phrases: isometric immersion, indicatrix, curvature, asymptotic line.

MSC: 53C23, 53C45

Received: 30.04.2020

Language: English

DOI: 10.15407/mag16.03.208

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