I have the following question:

Let $M\subset R^3$ be a regular surface and $A\in M$ a point where the gauss curvature $K$ and the mean curvature $H$ satisfy $K\neq 0, H=0$. Also, let $\alpha, \beta:(-1,1)\to M$ be regular curves such that $\alpha(0)=\beta(0)=A$ and the angle between their tangents at this point is not $0$. Also, $\alpha''(0),\beta''(0)\in T_AM$. Find the angle between the curves in $A$.

I can conclude from the condition on the curvatures that $K<0$. I am also familiar with the first and second fundamental forms, sectional curvature and curvature operator and "theorema egregium", Though I can't seem to find a direction to a solution.

Thanks in advance.


HINT: The conditions you were given on the second derivatives tell you what the normal curvatures of the curves are at $A$. There is a famous formula for the normal curvature in the direction $\theta$ relative to the principal directions; use it! (This question has appeared numerous times on this site, once you realize there's a word for these curves ...)

  • $\begingroup$ Could you please refer me to this formula? I don't think I know which one you are referring to. $\endgroup$ Feb 24 '20 at 7:03
  • $\begingroup$ Euler's formula. $\endgroup$ Feb 24 '20 at 7:20
  • $\begingroup$ Thank you, will look into that! $\endgroup$ Feb 24 '20 at 7:26

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