Past exam of the geometry course of the University of Oxford / 2008 / A / 2

For a regular parametrization choose the unit normal $n=(r_u\times r_v)/|r_u\times r_v|$. The first fundamental form is the squared length induced on tangent vectors:The second fundamental form records their normal bending:The Gaussian curvature isRegularity ensures $EG-F^2>0$.

Expanding the characteristic determinant givesThe product and sum of its two roots therefore giveReversing the normal changes both principal curvatures and the mean curvature in sign, while leaving their product unchanged.

Put $b={\gamma }^{\prime }+v{a}^{\prime }$. Then $r_u=b$ and $r_v=a$, soThis is the induced metric of the ruled surface wherever $b\times a\ne 0$.

For the ruled surface, $r_{vv}=0$, so $N=0$. AlsoThusThe denominator is positive at every regular point, proving that zero Gaussian curvature is equivalent to the vanishing of the stated scalar triple product. In particular the Gaussian curvature of a regular ruled surface is always nonpositive.

For the helicoid,Hence $E=1+v^2$, $F=0$, $G=1$. Since $r_{uu}=(-v\cos u,-v\sin u,0)$ and $r_{vv}=0$, both $L$ and $N$ vanish. The mean-curvature formula therefore givesFor comparison, $M=1/\sqrt{1+v^2}$, so $K=-1/(1+v^2{)}^2$; zero mean curvature does not mean zero Gaussian curvature.