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I have seen it stated that:

For every point in a surface of negative Gaussian curvature, there are exactly two asymptotic directions, i.e ones in which the normal curvature is zero.

How can this be proved? Moreover, can someone give an intuitive explanation of why this should be so?

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1 Answer 1

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The second fundamental form in this case is a symmetric bilinear form, which is non-degenerate, but indefinite, since its determinant is negative. Hence there are exactly two null directions in each tangent space. By definition of normal curvature, these are asymptotic directions.

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  • $\begingroup$ Thanks, though this is a bit much to wrap my head around. Is there an intuitive explanation? I have an image that explains why there are at least two such directions, but not one for exactly two. $\endgroup$ Apr 17, 2016 at 19:34
  • $\begingroup$ Observe that this is not a statement in geometry, but in linear algebra. You have a bilinear form $b$ on $\mathbb R^2$ such that $b(e_1,e_1)=r$, $b(e_2,e_2)=s$ and $b(e_1,e_2)=0$ with $r>0$ and $s<0$. Then $b(x,x)=x_1^2 r+x_2^2 s$ so the equation $b(x,x)=0$ boils down to $x_1/x_2=\pm\sqrt{-s/r}$, which gives two lines. $\endgroup$ Apr 18, 2016 at 9:48

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