On the Local Isometric Embedding in R³ of Surfaces with Zero Sets of Gaussian Curvature Forming Cusp Domains

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We study the problem of isometrically embedding a two-dimensional Riemannian manifold into Euclidean three-space. It is shown that if Gaussian curvature vanishes to finite order and its zero set consists of two smooth curves tangent at a point, then local sufficiently smooth isometric embedding exists.