TY - JOUR
T1 - The Semi-global Isometric Imbedding in <em>R</em><sup>3</sup> 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&amp;Pi] × [-&amp;delta, &amp;delta] where E, F, G are smooth functions of (u, v) and periodic in u with period 2&amp;Pi. Moneover K|_{v=0} = 0. K_r|_{v=0} ≠ 0. when&gt; 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.