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For any surface patch with the first fundamental form $$g(u,v) = \left[\begin{array}{cc} (c+a\cos{v})^2 & 0\\ 0 & a^2\end{array}\right]$$ The Gauss and Codazzi Equations are

\begin{align} ac\cos(v)+a^2\cos^2(v)-h_{11}h_{22}+h_{12}^2&=0\\ h_{11,v} - h_{12,u} + \frac{a\sin(v)}{c+a\cos{v}}h_{11}+\frac{\sin(v)(c+a\cos(v))}{a}h_{22}&=0\\ h_{22,u} - h_{12,v} + \frac{a\sin(v)}{c+a\cos{v}}h_{12}&=0 \end{align} If we can show that the solution for the functions $h_{ij}$ is the same as that for the torus patch $(c+a\cos(v))\cos(u), (c+a\cos(v))\sin(u), a\sin(v))$, then we are done by uniqueness part of the Fundamental Theorem of Surfaces (patches with the same $g$ and $h$ differ only by a rigid motion).

Remark: If we make an overly strong assumption that $h$ is diagonal then this gives the result, but otherwise, as Deane comments, it is not immediately clear how/if we can prove the uniqueness of the $h_{ij}$ in the general case.

Update: Consider a particular local isometry of a patch on the torus that is small enough to not create any umbillic points. We can reparametrise in the neighbourhood of any non-umbillic point to a principal patch where both $g$ and $h$ are diagonal. The first fundamental form for the reparametrised isometric patch will have the form

$$g(u,v) = \left[\begin{array}{cc} \lambda(u,v)^2(c+a\cos{v})^2 & 0\\ 0 & \mu(u,v)^2a^2\end{array}\right]$$

for known $\lambda,\mu$ and then the Codazzi equations are now a linear system for $h_{11},h_{22}$: $$h_{11,v} = \frac12\partial_v(\lambda^2(c+a\cos{v})^2)(\frac{h_{11}}{\lambda^2(c+a\cos{v})^2} + \frac{h_{22}}{\mu^2a^2})$$

$$h_{22,u} = \frac12\partial_u(\mu^2a^2)(\frac{h_{11}}{\lambda^2(c+a\cos{v})^2} + \frac{h_{22}}{\mu^2a^2})$$

and the Gauss equation is $$h_{11}h_{22} = \lambda^2\mu^2a\cos(v)(c+a\cos(v)).$$

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For any surface patch with the first fundamental form $$g(u,v) = \left[\begin{array}{cc} (c+a\cos{v})^2 & 0\\ 0 & a^2\end{array}\right]$$ The Gauss and Codazzi Equations are

\begin{align} ac\cos(v)+a^2\cos^2(v)-h_{11}h_{22}+h_{12}^2&=0\\ h_{11,v} - h_{12,u} + \frac{a\sin(v)}{c+a\cos{v}}h_{11}+\frac{\sin(v)(c+a\cos(v))}{a}h_{22}&=0\\ h_{22,u} - h_{12,v} + \frac{a\sin(v)}{c+a\cos{v}}h_{12}&=0 \end{align} If we can show that the solution for the functions $h_{ij}$ is the same as that for the torus patch $(c+a\cos(v))\cos(u), (c+a\cos(v))\sin(u), a\sin(v))$, then we are done by uniqueness part of the Fundamental Theorem of Surfaces (patches with the same $g$ and $h$ differ only by a rigid motion).

Remark: a change of variables to $h_{11}, h_{22},$ and If we make an overly strong assumption that $\det{h}$ should provide h$is diagonal then this gives the necessary simplification of result, but otherwise, as Deane comments, it is not immediately clear how/if we can prove the system uniqueness of equationsthe$h_{ij}in the general case. 6 added 17 characters in body For any surface patch with the first fundamental form $$g(u,v) = \left[\begin{array}{cc} (c+a\cos{v})^2 & 0\\ 0 & a^2\end{array}\right]$$ The Gauss and Codazzi Equations are \begin{align} ac\cos(v)+a^2\cos^2(v)-h_{11}h_{22}+h_{12}^2&=0\\ h_{11,v} - h_{12,u} + \frac{a\sin(v)}{c+a\cos{v}}h_{11}+\frac{\sin(v)(c+a\cos(v))}{a}h_{22}&=0\\ h_{22,u} - h_{12,v} + \frac{a\sin(v)}{c+a\cos{v}}h_{12}&=0 \end{align} If we can show that the solution for the functionsh_{ij}$is the same as that for the torus patch$(c+a\cos(v))\cos(u), (c+a\cos(v))\sin(u), a\sin(v))$, then we are done by uniqueness part of the Fundamental Theorem of Surfaces (patches with the same$g$and$h$differ only by a rigid motion). Remark: a change of variables to$h_{11}, h_{22},$and$\det{h}\$ should provide the necessary simplification of the system of equations.

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