I have a two part question, one of which I have no clue of how to approach, the other has been giving me a difficult time for the last three days. The questions are as follow:
Le M be an arbitrairy orientable surface, with the unit normal vector. Define the following map: by
i) Show that the canonical isomorphisms of make a unit normal on for which , where are vectors, and indicates the shape operator.
I have no idea what's being asked here.
ii)Derive the following formulas of the Gaussian and Mean Curvature
where and denote the principal vectors
Before I begin I managed to deduce from a previous problem that .
Now I would imagine that since that I would simply have wich would simply give which isn't the desired result.
So looking at both RHS and LHS I think I would get:
then from the RHS I get :
which has lead nowhere.
I'm not looking for the answer, but any hints would be greatly appreciated.