I'd like a hint for the following question: Show that in a elliptic point, principal directions bisects asymptotic directions. thanks.
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1How do you define "elliptic points", "principal directions", and "asymptotic directions"? What book are you using to learn differential geometry? – J. M. ain't a mathematician Nov 06 '11 at 09:39
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I'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. – Jr. Nov 07 '11 at 02:00
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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.
