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