On Isometric Immersions of the Lobachevsky
Plane into 4-Dimensional Euclidean Space
with Flat Normal Connection
B. Verkin Institute for Low Temperature Physics and Engineering of the National
Academy of Sciences of Ukraine, 47 Nauky Ave., Kharkiv, 61103, Ukraine
Received April 30, 2020.
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.