The Semi-global Isometric Imbedding in <em>R</em><sup>3</sup> of Two Dimensional Riemannian Manifolds with Gaussian Curvature Changing Sign Cleanly
Year: 1993
Journal of Partial Differential Equations, Vol. 6 (1993), Iss. 1 : pp. 62–79
Abstract
An abstract Riemannian metric ds²= Edu² + 2Fdudv + Gdv² is given in (u, v) ∈ [0, 2&Pi] × [-&delta, &delta] where E, F, G are smooth functions of (u, v) and periodic in u with period 2&Pi. Moneover K|_{v=0} = 0. K_r|_{v=0} ≠ 0. when> K is the Gaussian curvature. We imbed it semiglobally as the graph of a smooth surface x = x(u, v ), y = y(u, v), z = z(u, v) of R³ in the neighborhood of v = 0. In this paper we show that, if [K_rΓ²_{11}]_{v=0}, and three compatibility conditions are satisified, then there exists such an isometric imbedding.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/1993-JPDE-5700
Journal of Partial Differential Equations, Vol. 6 (1993), Iss. 1 : pp. 62–79
Published online: 1993-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 18