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If $u\in W_{\mathrm{loc}}^{1,2}$, then $Du=0$ almost everywhere on every level set $\{u=t\}$. Choose smooth truncations ${\Phi }_{\epsilon }(s)$ with $|{\Phi }_{\epsilon }|\le 2\epsilon$, derivative one near zero, bounded derivative, and derivative zero outside $(-2\epsilon ,2\epsilon )$. The Sobolev chain rule gives $D{\Phi }_{\epsilon }(u-t)\to 1_{\{u=t\}}Du$ in local $L^2$ by dominated convergence theorem, while ${\Phi }_{\epsilon }(u-t)\to 0$. The limiting weak derivative is therefore zero.

A hyperbolic circle of centre $o$ and radius $R>0$ is the level set $\{x:d_{\mathbb{H}}(o,x)=R\}$ for the hyperbolic distance. Equally spaced directions from its centre give vertices of a regular geodesic polygon.

A level curve is the planar level set of a scalar function. Where the gradient does not vanish, the implicit function theorem makes it locally a smooth curve. At a critical point, branches may cross, narrow into a cusp or shrink to a point, so a plot should be interpreted with the local critical-point analysis.

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.