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EXERCISE

a)Show that there is not surface with first fundamental form: $$ds^2=udu^2+vdv^2$$ that can be isometric with a circular cylinder

b)Find the Umbilical points of the elliptical surface:$$\frac{x^2}{2^2}+\frac{y^2}{3^2}+\frac{z^2}{5^2}=1$$

ATTEMPT

From the first Fundamental Form, we have that:

$E=u, G=v , F=0$.

We know that the parametrization of the circular cylinder is $(acost,asint)$

How can I continue so to show that $$E_p=u=E_{f_p}, G_p=v=G_{f_p}, F_p=0=F_{f_p}?$$

b)I don't have any idea how to work in this section of the exercise.

We know that a point is umbilical when we have that : $$FL-EM=GL-EN=GM-FN=0$$ or $$k=\frac{L}{E}=\frac{M}{F}=\frac{N}{G}$$

Our professor has never shown us a problem like this in class so I don't know how to work in these types of exercise.

I would very grateful if someone could help me with this, with any hint or a thorough solution and explanation!

Thanks, in advance!

Paris K. Patsogiannis
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  • For (b) you're going to have to compute the first and second fundamental forms. For (a) ask your professor if he means locally isometric or globally isometric. You can see that this metric is locally isometric to the usual flat metric on the plane, and the plane is, indeed, locally isometric to the cylinder. – Ted Shifrin Sep 16 '19 at 21:26
  • @TedShifrin But how can I compute the first and second fundamental forms only with the given type of elliptical surface? And thanks for all your help! – Paris K. Patsogiannis Sep 16 '19 at 21:48
  • This is an awful computation to do, but you can parametrize it by stretching spherical coordinates (just as you can parametrize an ellipse by starting with a circle and stretching it to get $(a\cos t,b\sin t)$). – Ted Shifrin Sep 16 '19 at 21:55

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