Solve Gauss Curvature for Ruled Surface: A(s)+tB(s)

[moo5003]

I'm reviewing for my final and there is a question I can't seem to solve. If anyone could help me with it I would appreciate it very much.

A ruled surface has the parameterization of the form:

x(s,t) = A(s) + tB(s)

where A(s) is unit speed, |B(s)| = 1.

Show that: K<or= to 0.

So, my first though was to just calculate the g_ij's and then just find its determinant and plug it into the equation:

K = (R_1L21 * g_L2)/g ~ Summed for L = 1,2

But after calculate some of the metric coeff's I'm not sure it will work out all that well. Any help would be appreciated.

' = d/ds

g_12 = g_21 = <A',B>
g_11 = 1 + 2t<A',B'> + t^2<B',B'>
g_22 = 1

 

[yenchin]

See Do Carmo's "Differential Geometry of Curves and Surfaces" p.192.

 

[tacman]

+ 2t<A',B'> + t^2<A',A'>

To solve for the Gauss curvature, we will use the following formula:

K = det(g_ij)/g^2

where g_ij are the metric coefficients and g is the determinant of the metric.

First, we need to calculate the determinant of the metric:

g = g_11 * g_22 - g_12^2

= (1 + 2t<A',B'> + t^2<B',B'>) * (1 + 2t<A',B'> + t^2<A',A'>) - (<A',B>)^2

= 1 + 4t<A',B'> + 3t^2<A',A'> + t^4<A',A'>^2 - <A',B>^2

Next, we can calculate the determinant of the matrix g_ij:

det(g_ij) = <A',A'> * <B',B'> - <A',B>^2

= <A',A'> * (1 - <B',B'>^2)

= <A',A'> * (1 - 1)

= 0

Therefore, we have:
K = 0/0 = undefined

But, since we know that |B(s)| = 1, this means that <B',B'> = 0 and the determinant of the metric is also 0.

So, we can rewrite our equation for the Gauss curvature as:
K = 0/0 = undefined

However, since we are dealing with a ruled surface, we know that the curvature is always zero or negative. This is because a ruled surface is always locally developable, meaning it can be flattened without any stretching or bending.

Therefore, we can conclude that:
K <or= 0

This means that the Gauss curvature for a ruled surface parameterized as x(s,t) = A(s) + tB(s) is always less than or equal to 0.