Volume 6, Issue 1
The Semi-global Isometric Imbedding in R3 of Two Dimensional Riemannian Manifolds with Gaussian Curvature Changing Sign Cleanly

Dong Guangchang

J. Part. Diff. Eq., 6 (1993), pp. 62-79.

Published online: 1993-06

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  • 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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@Article{JPDE-6-62, author = {}, title = {The Semi-global Isometric Imbedding in R3 of Two Dimensional Riemannian Manifolds with Gaussian Curvature Changing Sign Cleanly}, journal = {Journal of Partial Differential Equations}, year = {1993}, volume = {6}, number = {1}, pages = {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.}, issn = {2079-732X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jpde/5700.html} }
TY - JOUR T1 - The Semi-global Isometric Imbedding in R3 of Two Dimensional Riemannian Manifolds with Gaussian Curvature Changing Sign Cleanly JO - Journal of Partial Differential Equations VL - 1 SP - 62 EP - 79 PY - 1993 DA - 1993/06 SN - 6 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jpde/5700.html KW - AB - 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.
Dong Guangchang. (1970). The Semi-global Isometric Imbedding in R3 of Two Dimensional Riemannian Manifolds with Gaussian Curvature Changing Sign Cleanly. Journal of Partial Differential Equations. 6 (1). 62-79. doi:
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