= Constant Gaussian curvature surfaces of revolution
For an arc-length profile with radius $Y(u)>0$, the equation $K=c$ is
$$
Y''+cY=0,
\qquad
Z'=\sqrt{1-Y'^2}.
$$
Thus $Y=A\cos u$ gives $K=1$ locally, while $Y=A\cosh u$ gives $K=-1$ wherever $|Y'|<1$. At $u=0$ their mean curvatures are respectively
$$
\frac12\left(A+\frac1A\right)
\quad\hbox{and}\quad
\frac12\left(-A+\frac1A\right),
$$
so different choices of $A$ give locally noncongruent surfaces with the same constant Gaussian curvature.
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