 Level-line curvature

At a regular planar level set of a twice differentiable function, its oriented curvature with normal $\nabla u/|\nabla u|$ is $\kappa =(\Delta u-\nabla u^T(D^{2}u)\nabla u/|\nabla u{|}^2)/|\nabla u|$. Zero curvature at a point gives second-order contact with the tangent line, not a whole straight segment. For example the regular zero level line of $u(x,y)=y+x^3$ has zero curvature at the origin but is curved nearby.