I'd like a hint for the following question: Show that in a elliptic point, principal directions bisects asymptotic directions. thanks.
Question about asymptotic direction
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differential-geometry
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0I'm using Manfredo's book(Portuguese version) but there is a english version of it.Elliptic points are points whose Gaussian curvature is always positive, principal direction is a line generated by the eigenvectors of the self-adjoint transformation $DN_p$, asymptotic direction is a line generated by a vector which is tangent to a curve whose normal curvature is zero.I tried to use Euler's formula but get no result. – 2011-11-07
1 Answers
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Elliptic points do not have asymptotic directions, because the principal directions point in the directions in which the normal curvature obtains its maximum $k_1$ and minimum $k_2$. The curvature of a surface is $K=(k_1)(k_2)$. So if at an elliptic ($K>0$) point you have a direction for which the normal curvature is zero then $k_2$ must be zero, and you arrive to a contradiction.