# Differential geometry question

• Nov 18th 2007, 07:11 AM
musicmental85
Differential geometry question
Hi,
I have lots of theory to work off but no examples so I cant figure out
what Im doing for this question

Consider a unit-radius cylinder with centre running along the x-axis
a) write down a coordinate patch which covers this cylinder
b) compute the shape operater
c) find the principal curvatures and vectors
d) find the Gaussian and mean curvatures

I know one of the coordinate patches we have studied is the mange
patch. How do I know if that is a viable option here?

If it isn't I have a lemma that states for a coordinate patch PHI of a
function f:M->R we have PHI_u(f)=df/du and PHI_v(f)=df/du. Does this
mean I need to write a function for the cylinder and differentiate it
to get a patch?

The shape operater is S_p(v)=-(TRIANGLE)_v. n where n(p)=PHI_u(p) X
PHI_v(p)/||PHI_u(p) X PHI_v(p) but as I cannot work out what the PHI
is supposed to be I am completely stuck at the moment and can work no
further:confused:
• Nov 30th 2007, 05:12 PM
Rebesques
Let's see...

For a coordinate patch try $C: \Phi(u,v)=(v,cosu,sinu), u\in[0,2\pi], v>0$. Now we can calculate the normal at the point $p=\Phi(u,v)$, $\eta(u,v)=\frac{1}{(\sqrt{1+v^2})}\left(0,cosu,sin u\right)=(\eta_1(u,v),\eta_2(u,v),\eta_3(u,v))$.

For $w=(w_1,w_2)\in T_p(C)$, we have $S_p(w)=-\nabla_w(\eta)=-\left((d\eta_1)\Big{|}_{(u,v)}(w),(d\eta_2)\Big{|} _{(u,v)}(w),(d\eta_3)\Big{|}_{(u,v)}(w)\right)$
if my memory is not failing me again :o. For the first one say, we have

$(d\eta_1)\Big{|}_{(u,v)}(w)=\frac{\partial \eta_1}{\partial u}w_1+\frac{\partial \eta_1}{\partial v}w_2$, and so on.

The principal curvatures are just the eigenvalues of $S$.

For the Gaussian and mean curvature, either use the known formulae that include the principal curvatures, or play smart and notice that the cyhlinder and the plane are isometric, so...
• Dec 3rd 2007, 07:44 AM
musicmental85
Ok so I can work out the actual question when given the patch but how do I pick a coordinate patch?
• Dec 7th 2007, 07:50 AM
Rebesques
By force or by experience. :)

For the cylinder, it is quite easy, as two variables are always on a circle for fixed height. So you get polar coordinates to express these.

Why don't you try giving parametrizations for the sphere and the torus? Should be good practice.