Given some regular surface $S$ that contains a line $L$, I need to prove that the Gaussian curvature $K\leq 0$ at all points of $L$. I am thinking that if I could show that no points on $L$ can be elliptical $K>0$ then I would be set. But how can I do that?
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2Related to this question: http://math.stackexchange.com/questions/945762/gaussian-curvature-k-0?rq=1 – bubba Mar 21 '16 at 15:31
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Hint: The Gaussian curvature is the product of the principal curvatures. What do the principal curvatures minimize/maximize?
treble
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Since the normal curvature $k_n = kcos@$ and the curvature $k$ of a line is zero, then the normal curvature is 0. Then using the Euler formula, $k_n = k_1 cos^2(@) + k_2 sin^2(@)$, this shows that $k_1$ and $k_2$ have opposite signs, or are equal to zero. Since the Gaussian curvature $K = k_1 * k_2$, $K$ is less or equal to zero.
Silvia Rossi
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