 Solution

ID: past-exam-of-the-geometry-course-of-the-university-of-oxford/2008/a/2/iii/b/solution

r_{vv} = 0

, so

N = 0

. Also

M = \frac{( b \times a ) \cdot a ^{'}}{∣ b \times a ∣} = \frac{γ ^{'} \cdot ( a \times a ^{'} )}{∣ b \times a ∣}, EG - F^{2} = ∣ b \times a ∣^{2} .

Thus

K = - \frac{[ γ ^{'} \cdot ( a \times a ^{'} ) ] ^{2}}{∣ b \times a ∣ ^{4}} .

The 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.

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