First fundamental form of an arc-length surface of revolution

For $X(u,v)=(Y(u)\cos v,Y(u)\sin v,Z(u))$ with $Y^{\prime 2}+Z^{\prime 2}=1$, the induced metric is

With unit normal $(-Z^{\prime }\cos v,-Z^{\prime }\sin v,Y^{\prime })$, the principal-form coefficients give

For an arc-length profile with radius $Y(u)>0$, the equation $K=c$ isThus $Y=A\cos u$ gives $K=1$ locally, while $Y=A\cosh u$ gives $K=-1$ wherever $|Y^{\prime }|<1$. At $u=0$ their mean curvatures are respectivelyso different choices of $A$ give locally noncongruent surfaces with the same constant Gaussian curvature.

For $0<A<1$, rotate the unit-speed profilenear $s=0$. Since $K=-Y^{\prime \prime }/Y$, the resulting regular surface of revolution has $K=1$. At $s=0$ its principal curvatures are $A$ and $A^{-1}$, so the point is not umbilical when $A\ne 1$ and no neighbourhood of it lies on a sphere.

For the parametrizationthe Gaussian curvature and area element are

For $a\le x\le b$, the curvature formula is an exact derivative and gives