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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.

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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.

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  • $\begingroup$ Hello, good afternoon, @TedShifrin, I was able to work out the dot products, but I am unsure about the cross products. $f_u \times f_v = (c' + vX') \times X = c' \times X - v(X \times X')$ Am I doing the right thing? $\endgroup$ – James Nov 7 '17 at 19:49
  • $\begingroup$ I am done! It took a while to compute $\langle f_u \times f_v, f_{uu} \rangle$. $\endgroup$ – James Nov 8 '17 at 13:22
  • $\begingroup$ Yup, that one is quite involved, but the $f_{uv}\cdot\mathbf n$ comes into the $H$ computation, as well, because of the $I^{-1}II$ multiplication. $\endgroup$ – Ted Shifrin Nov 8 '17 at 16:43
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You already have written the FFF, to compute the normal vector you can normalize $ \tilde{N}=f_u \times f_v $. Once you get the normal vector $N = \frac {\tilde{N}} {|| \tilde{N} ||}$ you can compute the coefficient of the SFF as $\langle N, f_{x_i, x_j}\rangle$.

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  • $\begingroup$ Yeah, that's how you compute for $N$, but since $f_u$ and $f_v$ aren't expressed as components, how do you do it then? $\endgroup$ – James Nov 5 '17 at 18:29
  • $\begingroup$ If $\lambda\neq 0$ or $J\neq 0$ you have an orthonormal frame, obtained completing two vectors you know are already orthonormal for example $X,X'$ or $X',X''$, in determining the components along this new orthonormal frame the symbols $\lambda,J ,F$ will come into play. $\endgroup$ – Warlock of Firetop Mountain Nov 5 '17 at 18:43
  • $\begingroup$ Good day, I still don't understand how to do it. I don't know how to obtain the expression for $\langle f_u, f_u \rangle$, $\langle f_u, f_v \rangle$, $\langle f_v, f_v \rangle$. I don't know how to obtain the expression for $f_u \times f_u$. If I only knew, I wouldn't be asking. I need help, please. Thanks. $\endgroup$ – James Nov 6 '17 at 13:09
  • $\begingroup$ @James: Have you tried working out the various things in my hint? You do need to know basics of vector algebra and calculus, however. In particular, the cross product of any vector with itself is $\vec 0$. But you'll need the product rule for dot and cross products, among other things. $\endgroup$ – Ted Shifrin Nov 7 '17 at 16:29
  • $\begingroup$ Hello, good afternoon, @TedShifrin, I was able to work out the dot products, but I am unsure about the cross products. $f_u \times f_v = (c' + vX') \times X = c' \times X - v(X \times X')$ Am I doing the right thing? $\endgroup$ – James Nov 7 '17 at 19:48

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