Showing that an isometric deformation transforms the surface normals at corresponding points. I am basically trying to see that if given an isometric deformation $\Phi$ between two surfaces i.e.
\begin{equation}
\Phi : \boldsymbol{x}(u,v) \to \boldsymbol{y}(u,v)
\end{equation}
the normal vector $\boldsymbol{n}_x(u_0,v_0)$ transforms into $\boldsymbol{n}_y(u_0,v_0) = \Phi(\boldsymbol{n}_x(u_0,v_0))$.  Can this be proved?? I would like to clarify that an isometric deformation is slightly stronger than an isometry requirement as it implies a continuous bending as explained in this link.
An obvious example of this is the folding of a flat sheet of paper into a cylinder or the deformation between a catenoid and a right helicoid. A non-example would be a right circular helicoid and a left circular helicoid which are locally isometric.
 A: $\newcommand{\Eucl}{\mathbf{E}}\newcommand{\eps}{\varepsilon}$The idea of a continuous isometric deformation can be made precise as a path in the space of local isometries, but can also be formulated in more concrete terms:
Let $U$ be an open subset of the plane and let $\Phi_{0}:U \to \Eucl^{3}$ be a regular parametrization of a (possibly immersed) surface $S_{0}$ in Euclidean three-space. We can define an isometric deformation of $S_{0}$ to be a smooth mapping $\Phi: U \times (-\eps, \eps) \to \Eucl^{3}$ (for some real $\eps > 0$) such that

*

*$\Phi(u, v, 0) = \Phi_{0}(u, v)$ for all $(u, v)$ in $U$;

*For each $t$ in $(-\eps, \eps)$, the restriction $\Phi_{t} = \Phi(\cdot, \cdot, t):U \to \Eucl^{3}$ is a regular parametrization of a surface $S_{t}$;

*Let $g$ denote the Euclidean metric. For each $t$ in $(-\eps, \eps)$, the metric $\Phi_{t}^{*}g$ is equal to $\Phi_{0}^{*}g$. (This condition can be expressed in terms of dot products of the partials of $\Phi_{t}$, and implies that if $\Phi(u, v, 0) \mapsto \Phi(u, v, t)$ defines a mapping $S_{0} \to S_{t}$, then this mapping is a local isometry. We state things in this roundabout way because, for example, we might want to view a helicoid as a smooth deformation of a catenoid, see also below.)

For such a deformation, the unit normal fields "vary continuously" because they can be calculated in terms of the partials of $\Phi_{t}$.
Incidentally, as noted in the comments, helicoids of opposite chirality actually are deformable into each other (either locally, or globally through immersions), e.g., by
$$
\Phi(u, v, t) = \cos t (\cosh v \sin u, -\cosh v\cos u, -v) - \sin t(\sinh v\cos u, \sinh v\sin u, u).
$$
(At $t = \pi/2$ the helicoid is right-handed; at $t = -\pi/2$ it is left-handed.)
