Determining a surface of revolution from the metric function

Question

Suppose we have a surface $X(u,v)$ where $(u,v)$ are conformal curvature line coordinates with the first fundamental form $$ds^2 = \lambda^2(du^2 + dv^2).$$

Does the fact that $\lambda_u = 0$ (or $\lambda_v = 0$) imply that this surface is a surface of revolution?

Answer 1

Suppose the induced metric of an embedded surface $S$ has the form $ds^{2} = \lambda^{2}(du^{2} + dv^{2})$ for some function $\lambda$ of one variable, and the coordinate curves are principal, i.e., lines of curvature. Does this imply $S$ is a surface of revolution?

No; $S$ could be a cylinder (over an arbitrary curve) parametrized by $$ X(u, v) = (x(u), y(u), v),\qquad x'(u)^{2} + y'(u)^{2} = 1. $$