A surface F2 in E4 is called function-degenerate if parameters α, β, a, and b of the ellipse of normal curvature (α and β are the coordinates of its center in the normal plane and a and b are the values of the semiaxes) are some functions of a function t(P) of point P of the surface. This class of surfaces is a generalization of the class of surfaces that admit motion with respect to themselves along some family of lines. The following theorem is proved: There does not exist a regular, class C6, isometric immersion of the Lobachevsky plane L2 into E4, with zero Gaussian torsion, in the form of a function-degenerate surface. This theorem includes the Hilbert theorem on surfaces of negative curvature in E3.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Yu. A. Aminov, “On immersions of n-dimensional Lobachevsky space into 2n-dimensional Euclidean space with n fields of principal directions”, Ukr. Geom. Sb., No. 28, 3–8 (1985).
S. B. Kadomtsev, “Investigation of uniqueness problems of two-dimensional surfaces in Euclidean spaces”, Probl. Geometrii,8, 243–256 (1977).
E. P. Rozendorn, “On the problem of immersion of two-dimensional Riemannian metrics in four-dimensional Euclidean space”, Vestn. Mosk. Univ. Ser. I Mat. Mekh., No. 2, 47–50 (1979).
Yu. A. Aminov, “Isometric immersions, with flat normal connections, of domains on n-dimensional Lobachevsky space in Euclidean spaces. A model of a gauge field”, Mat. Sb.,137. No. 3, 275–298 (1988).
Translated from Ukrainskii Geometricheskii Sbornik, No. 33, pp. 8–18, 1990.
About this article
Cite this article
Aminov, Y.A. On function-degenerate immersions of a Lobachevsky plane into E4 . J Math Sci 53, 467–474 (1991). https://doi.org/10.1007/BF01109644
- Normal Curvature
- Negative Curvature
- Isometric Immersion
- Normal Plane
- Lobachevsky Plane