We are asked to prove the following theorem found in page 88 of Differential Geometry: Curves, Surfaces, Manifolds by Wolfgang Kühnel.
Using standard parameters, calculate the Gaussian curvature and the mean curvature of a ruled surface as follows:
$K = -\dfrac {{\lambda}^2} {{{\lambda}^2 +v^2}^2}$
and
$H = -\dfrac {1} {2({\lambda}^2 +v^2)^{3/2}} (Jv^2 + \lambda' v + \lambda (\lambda J + F))$
In standard parameters, a ruled surface is $f(u,v) = c(u) + v X(u)$ and $||X|| = ||X'|| = 1$ and $\langle c', X' \rangle = 0$.
Thus, using standard parameters, a ruled surface is, up to Euclidean motions, uniquely determined by the following quantities:
$F = \langle c', X\rangle$
$\lambda = \langle c' \times X, X' \rangle = \det (c', X, X')$
$J = \langle X'', X \times X' \rangle = \det (X, X', X'')$
Also, the first fundamental form is given as follows:
$I = \begin {pmatrix} \langle c',c' \rangle + v^2 & \langle c', X \rangle \\ \langle c', X \rangle & 1 \end {pmatrix} = \begin {pmatrix} F^2 + {\lambda}^2 + v^2 & F \\ F & 1 \end {pmatrix}$ with $\det (I) = \lambda^2 + v^2$.
So far, I have that
$f_u (u,v) = c' + vX'$
and
$f_v (u,v) = X$
Also,
$f_{vv} (u,v) =0$
I know the formula for the first fundamental form, the normal vector, and the second fundamental form. However, I don't know how to obtain the dot products and the cross products without having to isolate the components.
Best Answer
HINTS: The Gaussian curvature is clear, since we can take $\det(II) = -\lambda^2/(\lambda^2+v^2)$ and divide by $\det(I)$. [Start by showing that $\det(I) = \|\mathbf n\|^2 = \lambda^2+v^2$.]
You just need to work it all out carefully, using all the information you have. For example, $\langle X'',X\rangle = -1$ (why?). And because $\langle c',X'\rangle = 0$, we know that $\langle c'',X'\rangle = -\langle c',X''\rangle$. And we know, for example, that $c'\times X = \pm\lambda X'$ (why?). Calculating $H$ is a bit trickier, as you need to multiply $I^{-1}II$ before you can take the trace. $f_{uu}\cdot\mathbf n$ will have about 5 terms in it, for example ... Have fun and keep me posted.