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Do carmo: 3.2 E1- Hyperbolic point and hyperdirection

Barioth

Member
Jan 17, 2013
52
Hi everyone, I'm struggling a lot with this book, I have this problem that I don't know how to tackle,

its from Do Carmo, Chap(3.2) E1:
Show that at a hyperbolic point, the principal directions bisect the asymptotic directions.


If you google the book name, the first link is the PDF version of it, I leave it here, if anyone want a look at it.
(My problem is at page 151)
http://www.dic.unipi.it/davide.tonelli/files/dC.pdf
 

Barioth

Member
Jan 17, 2013
52
I'm still working on it, reading trough the section again, I'm having some problem with the notations used, maybe someone is familliar with them,

we have the gauss map N.

We know that for a Surface $S$ and a point $p \in S$
the Gaussian curvature $K$ of $S$ at $p$ is given by the determinant of $dN_p$.

Now my problem is that from my understanding N is given by
\(\displaystyle N(p) = \frac{x_u \bigwedge x_v}{|x_u \bigwedge x_v|} (p)\)

with x(u,v) a parametrization of a regular surface S.

From what I understand,$dN_p$ is given by the matrix $(N'_u(p),N'_v(p))$ which is a 3x2 matrix. Which as to be wrong or else the determinant of \(\displaystyle dN_p\) is not defined. (since the matrix isn't squared)

Which make me realise, that one of my definition is wrong, I believe its the one of $dN_p$, since for it to make sense with the definition of the Gauss curvature, it should be 2x2.

edit: from further reading, I would guess that

\(\displaystyle dN_p= \left(\begin{matrix}a&b\\c&d\end{matrix}\right)= -\left(\begin{matrix}E&F\\F&G\end{matrix}\right)^{-1}\left(\begin{matrix}L&M\\M&N\end{matrix}\right)\)
where E,F,G are the first fondamental form and L,M,N are the second fondamental form. (In do cormo he use e,f,g for the second fondamental form, but on the internet it seems that L,M,N is more used)

Is there another way to comptuer $dN_p$ then computing the first and second fondamental form, then using these matrix?

edit: ( again =])

I might have an idea on how to solve this,

let $e_1$ and $e_2$ been the pincipal direction, at $p$ and hyperbolic point. We need to show that the principal direction bisect the asymptotic direction.

let $O$ being the angle beetween $e_1$ and a direction (v) on the tangent plane.

using the euler formula we have that

\(\displaystyle K_v=k_1*Cos²(O)+k_2*Sin²(O)\)

now for v to be an asymptotic direction we need that $k_v = 0$, but since p is an hyperbolic point, we have that $k_1 > 0 , k_2 < 0$

then
\(\displaystyle 0=k_1*Cos²(O)+k_2*Sin²(O)\)
\(\displaystyle k_1*Cos²(O)=-k_2*Sin²(O)\)
\(\displaystyle Tan²(O)= \frac{-k_1}{k_2}\)

we have that $O$ has 4 solution in $[- \pi, \pi]$ which are: $O, -O, -\pi+O $and $\pi - O$

which make some sort of cross in the trigonemetric circle, therefor we can conclude that the principal direction bisect the asymptotic direction.

Is this good? (Poolparty)
 
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