Parabolic points and curvature.

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I have problems to solve this exercise: Let $p$ be a point of an oriented surface $S$ and assume that there is a neighborhood $U$ of $p$ in $S$ all points of which are parabolic. Prove that the (unique) asymptotic curve through $p$ is an open segment of a straight line.

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1 Answer

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This is a nice question. Here's a hint: Assume you have a parametrization where the $u$- and $v$-curves are lines of curvature. Apply the Mainardi-Codazzi equations to deduce that an appropriate Christoffel symbol vanishes. This, in turn, will tell you that the parameter curve is a line segment.

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