1. ## 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

2. Let's see...

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

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

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

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

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...

3. Ok so I can work out the actual question when given the patch but how do I pick a coordinate patch?

4. 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.