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: